It's very typical for a school to discourage parents from seeking a diagnosis for a child who is not disruptive.
As you've discovered, undiagnosed kids get so far and then it does become a problem for them. There's no single age when this happens, for some it's starting school or earlier, for others it can be into middle age. The one common denominator is they manage okay until they don't. Well done for spotting your DD is struggling with social communication.
How open is she to the idea she's autistic? That can play a large part in the diagnostic procedure. Girls and women present differently to boys and men and most diagnostics are very male-based.
The NHS route is a long one, pre-covid waiting lists were 18 months to 2 years in many areas, but don't be put off, do check with your GP. see/phone chat him/her on your own initially with as many examples as you can of your DD's behaviour that's different to her peers to outline your concerns and ask for a referral.
Otherwise, you can have a private assessment, but make sure whoever does it also diagnoses for the NHS as some local authorities refuse to acknowledge private diagnoses.
Lorna Wing is the Gold Standard, others can vary quite a bit, some including LW do what's called a 'pre-diagnostic' which will let you know if it's likely autism could be applicable. You've already outlined enough in your post to suggest it's a possibility. There are a lot of online questionnaires that can do that for free, e.g. the AQ test.
There are lot of channels on youtube by autistic young people, if she's not already watching any, perhaps research some yourself and if you think she'd enjoy them, watch them together. There are a good many about autistic anxiety.
As a parent, I like to watch Purple Ella, she can verbalise things about being autistic that my DD can't so it helps with my understanding.
You will notice that the more anxious your DD is, the more you notice her autism. If you can keep the anxiety at manageable levels, everything else seems to subside, at least that's how it is in our house.
The NAS have revamped their website so everything I could have linked to easily for you is now somewhere else, try this if you haven't already.
www.autism.org.uk/directory.
Autism terminology often changes. Asperger's has been omitted from the diagnostics because Hans Asperger had a murky past, nowadays the diagnosis tends to be ASC or ASD. C for condition replacing D for disorder.
Diagnosis is a complex process, it's not someone just picking a term out of thin air after reading a parent's description. I've no idea how it's carried out during lockdown but pre-Covid a child would be seen by a team of medics, usually Paed, Salt, OT and Ed Psych in at least 2 different settings, then the group would discuss their findings and prepare a comprehensive diagnostic report of attributes, deficits and recommendations.
That's a very longwinded way of answering Yes to your question "Is it worth trying to seek assessment"