DH and I were both privately educated - from 7+ as a boarder from overseas (DH) and from 11+ as a boarder with 100% scholarship (me).
Our children, one just left school, one in Year 12, have been state-educated throughout. Like you, we live in a place where these local (non-selective) schools are very good indeed. We could, just about, have funded private education - or followed SiL's lead, where FiL pays all fees for her 3 children - but chose not to.
DS is a musician, now at conservatoire, while DD is likely to pursue a more mainstream academic path post school. Both have very good GCSEs (in DD's case, excellent ones). DS has really good A-level grades 'to fall back on'. Both have led rich and varied extra-curricular lives, at community provision that in many cases exceed in standard those that could be provided by any one individual school. Both mix with anyone, can take a wide range of views and needs into account when considering any question affecting 'the wider society', and have a deep, though subtle, self-confidence that is the opposite of brash.
Yes, they might have acquired a little more obvious 'polish' elsewhere. Yes, they might show their self-confidence in a more overt way. Both might have achieved maybe 1 higher grade in 1 GCSE.
Do we regret our choice? No. It is also always worth remembering that, except in the case of the most extraordinarily selective schools, the route in at a slightly later date is almost always possible. The route back into state is much less easy.
HOWEVER, it does depend critically on the specific schools you are comparing - locally, the private options are not (with 1 exception) notably better / more academic than state.