I used to be very anti stop bath and still am with film - I have had pin holes that were caused when the ph basic developer engaged the acid stop bath in the emulsion. Does development continue in a water bath? - absolutely - until it is exhausted. I know that exhaustion of developer in the emulsion could take as much as two minutes in a water bath - or longer. Is this important? I don't think so - I usually rinse my film for 2 or more minutes in a water bath before going to the fix. I do NOT use acid fixer - only ammonium thiosulfate based with a ph of about 8. I test my fixer with every use. My fixer TF3, can clear film in 30 seconds. At 45 seconds to one minute of clearing time, I chuck it for fresh. I did not do this in the early years and my legacy work shows it. This is the error that comes back to haunt you in 5 years or so. Keep your chems fresh.
I did go back to using stop bath in paper though. It is not so important with normal paper developers (Like PC-TEA or Dektol) but with Amidol based (staining) developers, I noticed staining when I only used a water stop. I get this stop bath mix from B&H photo - it smells like vanilla. It is inexpensive, it has a traditional indicator in it and I have had no staining issues with Amidol since using it. I also know that my fixer lasts longer and I can switch the lights on a little quicker to examine a print when I use stop bath. I do not know what is in it but it is nothing like acetic acid. - Likely something more like citric acid. I guess there are issues now in shipping acetic acid so traditional stop baths are less available for mail order sales.
I also use a fixer clearing agent - I do not do the two bath fix routine - but I make my first wash one with some sulfite in it.