Hi Chris
I don't know of any problems as you describe with the BSA side valve engines, in fact they have a very good reputation for reliability, perhaps if anyone reading these posts knows differently they can advise.
As I have said previously the B series was developed from the M series and used the same (almost identical) bottom end, with just a longer con rod different barrel and head, also the M series continued for a few years alongside the B series, I don't think it likely that BSA would develop a flawed engine, more likely to start again.
Please don't get the wrong idea I am not trying to sway you to buy BSA, just trying to be constructive, as well as three BSAs I also have Triumph, Royal Enfield and Velocette bikes.
Incidentally all the B series are OHV.
Regards
Dennis