I don't know anything about contextual offers, but I assume that if you are close to the requirements for a standard offer of interview, then you would be a strong candidate on a contextual basis.
If you manage to get AAA predicted for A level and meet minimum academic requirements (which I assume you do), then your UKCAT score alone was above the standard cut off scores for interview at:
Sheffield - 672 last year (although SJT Band is used as a virtual MMI station so counts towards interview score)
Manchester - interview top third of UKCAT nationally so 7th decile (top 30%) fits that ok (667 last year).
Bristol - using UKCAT only for first time this year. They say that if they had done that last year the cut off would have been 665. Of course the profile of applicants could be different this year, particularly as they have changed their criteria
Newcastle - according to their website it was 645 last year, but 667 the year before
Southampton - historically on their open days they mention 640, but I couldn't find anybody on TSR last year with an interview below 667
Glasgow - 663 last year
Plymouth - last year 600, indicating 583 on their website for 2019 entry
St Georges - between 630-650 last 3 years
Also look at Liverpool - in their first sift they mark 9 x GCSE’s with 2 points for A*/A and 1 point for B grades. You need to achieve a score of 15 to be considered further. This process is advantageous to applicants with mostly A grade GCSE rather than A*, because they are treated the same in scoring terms.
https://www.liverpool.ac.uk/media/livacuk/schoolofmedicine/documents/Guidelines,for,applicants,direct,from,school,college,offering,GCSEs,Alevels,2019,entry.pdf Candidates are then ranked by UKCAT. The last three cut off scores have been 615 (last year), 640 and 625.
In terms of your Personal statement you seem to have strong examples to use for it. The only problem with the PS is that the assessment of it is very subjective, so how highly it's marked is unpredictable. Bristol, as mentioned above, are no longer PS heavy for standard entry. King's don't use PS in deciding on interview for standard entry.
UCL and ICL use BMAT for standard entry, not sure on contextual.
Sorry, I haven't been able to answer about the contextual aspect, but if you get predicted AAA there are plenty of options even if you applied for the standard routes. May be check out the contextual arrangements for some of those?