My wife is now lobbying me to buy a new TV I'm not sure that is the answer because I think that it is inadequate transmission power and I don't know who to contact to ask their opinion.
Subtitles disappearing is unlikely to be to do with transmission power.
Freeview uses digital transmission to carry sound, pictures and subtitles in the same digital data stream. If you get a good picture and good sound then your received signal will also receive good subtitles. If your picture doesn't break up into blocks and if your sound doesn't have disturbances (often sounding like chirps) then you have a decent signal.
If that is the case then, as others have suggested, your receiver could well be the issue. If possible borrow a Freeview set top box and try that connected to your aerial. If that solves your problem - then buying a cheap set top box (SD ones are often available for a tenner in supermarkets) might be a more cost effective solution than buying a new TV.
I suppose that the only way to guarantee subtitles is cable or satellite but I know nothing about either.
Freeview is perfectly reliable for subtitles as long as you have a reasonable signal and your receiver works properly.