I've done this. I trained in secondary MFL and worked as a secondary school teacher for five years before going on maternity leave. I never went back and set up as a private tutor instead.
I now do it full time (30 hours of teaching a week plus preparation and admin). I earned £40k last tax year and am hoping for nearer £45k this year. I charge £40 an hour for A level, £36 for everything else. I have 38 "official" teaching weeks a year that students are committed to but with school holiday teaching and extra lessons at exam time, it really adds up. I work after school (usually 3 or 4 hours), and all day Saturday and I do a small amount of day-time teaching (adults and university students and the occasional home-schooled child). I absolutely love it! I love working from home, I love the independence and I feel that I'm making a real difference to each individual student.
I started with French and German but then branched out into English and then Spanish (all four subjects at secondary - mostly GCSE and A level). After a while, I started to offer primary English and was then asked to do 11+ preparation, which I did. At first, I just did English, VR and creative writing for 11+ but the demand was so great that I started offering maths as well (plus NVR). (I do live in an 11+ area and there is huge demand.)
Pre-covid, I did mostly face-to-face teaching at my home, with some online students. All switched to online at the beginning of the first lockdown. I'm now starting to offer both again. It's almost always the primary parents who ask for face-to-face; the older ones are mostly happy online. I have never travelled to students' houses.
Things I've learnt:
Parents don't really ask about qualifications! Mine are all listed on my website, so maybe that's why, but not a single primary parent has ever queried that fact that I trained in secondary and have never taught in a primary school. I'm sure they do look for a qualified teacher though. (I also have absolutely no qualifications in Spanish but get loads of queries for Spanish and have tutored lots of children in it.)
You really need to know your stuff! It's so individual and so intense that you absolutely cannot wing it. (Not that I think you would!) If you're just teaching your specialism (in your case, primary maths and English?), then this shouldn't be a problem but if you branch out (which you may or may not need to, depending on demand and how many hours you want to do), you need to be absolutely sure of your ground, for your sake as much as theirs! (I actually offered free Spanish lessons to the daughter of a friend for a year to be sure I was up to scratch before unleashing myself on members of the public.) So you absolutely could branch out into secondary (maths or English or anything else) but you would need to be really sure of your subject knowledge first. For 11+/grammar school entrance, you also really need to know about the different schools and all the exam requirements.
You need to be really strict with parents or some will really mess you around in terms of cancelling lessons. I learnt this the hard way. I now send out invoices to be paid half-termly in advance. That takes up quite a bit of admin time but it's really worth it. If they can't make the lesson, I do offer to rearrange it, but either way, I don't lose the money. Nobody has ever expressed any dissatisfaction with the system.
I paid a professional to set up my website and I pay him an annual fee to run it. It's worth every penny and makes me look serious and professional. I used to use Google ads in September each year (I did that for about three years I think), which really helped to build things up. I don't need those now, as I am pretty much permanently fully booked (sometimes with a waiting list), but I would do it again in future if need be. I've never used an agency.
You need to keep really detailed records - financial of course but also for each child. In my case, it's what schools they're applying for for 11+, what exam board they're using for GCSE/A level and of course what work I've done with them.
Sorry - that was a bit of an essay. I really love it so much though and I think you should go for it if the maths adds up for you in terms of earnings. Bear in mind that it takes time to build things up, as I'm sure you realise. Feel free to PM me.