Can children get into selective secondaries without tutoring or holiday courses?

It’s increasingly unlikely if your kid is at a state primary. You would need to have a very special child who could be motivated to self study for the selective exams

The keywords are preparation, not tutoring or summer courses. For a selective school, you need thorough preparation; for a super selective school, you need extensive preparation. IMO, outside tutoring and summer course are commercial but not essential. The prerequisite, however, is that your child is already academically competent to begin with, and would have no problem doing off-the-shelf practice papers bought from the market without any preparation.

I teach English and prepare children for selective independent-school entrance and I wouldn't assume that a bright child necessarily needs intensive tutoring or a holiday course.
Where English is the weaker area, I would focus first on familiarity with the type of reading and writing they will actually be asked to do. The biggest gaps I tend to see are inference (rather than straightforward retrieval), explaining the effect of language rather than simply identifying techniques, and producing controlled creative writing within a time limit.
I would much rather see a child do one good practice paper, go through it carefully and understand how to improve it than race through lots of papers. Timed practice is useful too, particularly if their school hasn't prepared them for entrance exams.
So yes, DIY preparation can absolutely be worthwhile. I'd identify the specific English weaknesses first and spend the time and money there rather than feeling you have to sign up for everything on offer.

Woah 100 pounds? That's crazy. The one we got was from iTutorOnline and there was maybe 30 pounds per hour, no subscriptions or signup fees or anything, just paid straight for the hours. I think I ended up spending around 300 pounds in total and it was well worth it for us.

The issues are that most entrance exams are in the first term of year 6, if not earlier (one grammar near me has its exam next week) but the test expects a full knowledge and understanding of the year 6 maths curriculum and beyond. However bright the child is, someone needs to teach them that content. Then they not only need to know it but be able to apply it correctly and quickly under exam conditions. Unless, as a parent, you’re familiar and confident with year 6/7 maths then a tutor who is would be a wise choice.
Likewise, VR and NVR needs to be taught and practised. Being able to answer questions quickly is key.
I tutor for 11+. I can tell if a child has a chance or not by 4 things:

1. They need to working at the top end of greater depth in maths and preferably English too
2. They need to want to get in and be prepared to work hard, including completing homework weekly
3. They need to be reading a wide variety of books
4. The parents have to be on board to support with homework, vocabulary development and reading

Tutoring needs to start towards the end of Year 4 and be consistent. It needs to be a priority. Not cancelled in favour of sports and other hobbies. The tutor can be a professional or the parent.
All of the above is pertinent to my area. I appreciate it may differ in other areas.

It depends on the definition of "tutoring." If you mean hiring a private tutor, it’s definitely possible to pass without one - though it still takes a solid foundation and guided prep (whether parent-led or self-driven).
We didn't use a tutor because we weren't initially targeting super-selective schools; we felt our child was well on track for the selective ones we had in mind, so had light prep. It was only later on that we decided to try for some of the names you mentioned, and the outcome was better than expected. That said, had we aimed For the super-selectives from day one, peer pressure probably would have pushed us to get a tutor! (not necessary though)

I did, parents got some books from wh smiths a few weeks beforehand. Having said that it was less competitive in those days.

Honestly depends on your child, if they are in the top cohort at their current school then sure you can do it yourself, there are loads of prep workbooks available.

You definitely don’t need to pay a third party for tutoring unless you’re extremely time-poor and really can’t do it yourself, but it would be difficult for a child to get through with no prep whatsoever. I just used past papers and books from WH Smith and both DC got into fairly tricky ones.

From first hand experience, I would say there is no harm in it at all. When children enjoy learning, continue encouraging it. If a child enjoyed tennis, you wouldn't just stop them from playing or having extra sessions. Now, if you are not fussed on the school, you can just see how they get on. However, if you are keen on a particular school, it would be useful to maybe have a tutor to guide her in the techniques. I've found HB is harder to get into, especially as it is a grammar school. Even when working in a school teaching 1-2 years in advance, only the top 25% of the class would typically get in.

It's possible, but you're up against most who have been heavily tutored both for content and exam technique.

My gd is being tutored, the verbal reasoning is something she had not encountered (I had a go, ex teacher 2 post grad degrees, it was very difficult!)

Lots of practice papers is the best way anyway. Past papers can be printed out for free or at low cost

It really depends on the school, getting into a private can be easier then HB.

We paid £35 an hour for 1:1 in Oxfordshire. £100 is nuts!

It's competitive. It'll take a lot of effort, but having a tutor does make it more "guaranteed" even though that's not really a thing. More efficient and likely, for sure. I would say keep your eyes open at popular tutoring platforms like iTutorOnline, Superprof, Spires, etc.

Henrietta Barnet has 3,500 applicants for 120 spaces and if you imagine that many of those 3,500 will be in the top 10 of their current class, your daughter will be competing with people similar to her.

I agree that people who say they don't tutor for these types of schools are being dishonest. Likely the parent is going through books with them at home and to be honest this is the best way because then you know how they perform in the mocks. A tutor will never tell you this, they will always tell you how fab and smart your child is as they want to continue with you as a revenue stream.

Nearly everyone will improve their expected score on an exam if they get more familiar with the exam format, e.g., what do the questions look like and what are the timings on the sections. Doing past papers goes a long way towards this.

Nearly everyone will improve their expected score on an exam if they understand the marking scheme, e.g., which specific attributes of a writing piece are rewarded and which questions are worth more relative to the time required to answer them. This is where specialist help such as a prep school, a tutor, or a holiday course can be useful.

In answer to your specific question, it is very likely possible for some students to pass an exam with no preparation, but the chances that any individual student will pass is higher if they prepare. Especially since there are a fixed number of places so the passing score is determined by the competition and many of the competitors will have prepared.

It's not only that all-throughs want (most of) their students to stay. They sell it as a feature that, compared to preps, not needing to spend a half-year or more preparing for this specific set of exams means they can spend more time on more interesting and, possibly, useful learning.

They taught the kids about it post SATS! The school's overall results were fairly dismal!

I think you can tutor yourself. But they need to be well prepared for exam by going the past papers and books together.

Very occasionally but why risk it if its important to you?
The main thing they need is exam technique and learning how to answer the specific type of questions on the 11+.
A friend invigilates the exam at a local Grammar and she says that each year she sees children in tears because they have never seen anything like the exam. Some parents think any bright child will do well
DD did a load of exam papers and some group tuition for a couple of terms and did really well and some of her friends did similar. She did 1 summer exam with a 1 day session a week later to go over her mistakes but that was all

Mean and mode is on the year 6 curriculum so the primary school failed in their duty to teach it!

My DS had tutoring from yr 5 and got offers from 2 independent schools (one selective). Funnily enough in his Maths SAT exams he was the only person in his year to understand what mean and mode meant because he had done this with his tutor!

If you are not too bothered about which secondary he goes to then wing it but if you want to make sure 100% of offers from certain schools then definitely do the prep. It's really only for a short time compared to the next 5 or 7 years of the child's education.