It is impossible to know how thin your paint is without measuring it. Assuming someone has not bashed away the paint with some strong compounds, the paint should be ok. The flaking paint could be down to neglect or could indicate too much paint has been removed in the past.
If the car was mine I would give it a damn good washing and then use a paint cleaner, loads of these for sale eg Meguiars Stage 1 cleaner (from Halfords) or Serious Performance Paint Cleaner. Follow with some claying, again huge number of clays for sale although many now like the clay mitts (G3 from Halfords is fine). The reason for this approach is the paint will be chemically cleaned without removing any paint.
This should improve the gloss of the paint and at the very least will give you a better view of the paint. Products such as Autoglym's SRP are very good as they have a mild chemical cleaners and some very mild abrasives but will hide any minor marks. Nothing wrong with hiding defects, it all comes down to what you want to do. Maybe worth using something like SRP with a wax and wait until spring to work out what to do next.
As the outside temperatures are beginning to drop, many products (waxes) can be a bit tricky to apply although if you have a (heated) garage this should not be a problem. I use either a spray wax which is full of UV filters or spray sealant without problems in winter as a drying aid so they provide protection for the paint and make drying quicker.
Not sure how much machine polishing would cost, my guess is several hundred quid upwards although I would get a bodyshop to sort out the flaking paint first. If you do decide to use a professional detailer, do your research, there are some dodgy people out there as well as some very skilled workers. Machine polishing is not a hard skill to learn if you want to do it yourself but it is understandable if you wish to use a professional.