Fair point.
Anywhere you draw the line is arbitrary. 3 weeks, 3 months, 3 years... It may still attenuate lower depending on the conditions. The longer you wait the safer it is, but even beer that's been aging for years isn't guaranteed to be at FG. That's why traditional sours are frequently corked and caged or pasteurized/filtered.
It's smart to keep an eye on the bottles so they don't over-carb. If they do, you can simply put them in the fridge to stop or drastically slow additional fermentation.
If you'd monitored yours then perhaps you would have been able to save them.
For a contaminated beer, it's arguably better to bottle, carbonate, chill, and drink faster so that there's less chance for the contaminants to contribute flavor.
Plus, by the time you finish 2-3 months of additional aging you could have already finished drinking all the beer. If you want to age longer than that, maybe it'd be better to brew a batch that's not contaminated.
Waiting a reasonable amount of time (a few weeks at stable FG), good hydrometer reading skills (correcting for temperature, ideally using a higher resolution FG hydrometer), and most importantly,
paying attention to carbonation level after bottling means you'll be totally safe.