IIRC the B737-300's with the older analogue pressurisation controllers came with a customer option for the cabin rate controller. Either 500fpm or 350fpm in descent.
If you don't know the version of your aeroplane then assume it has the 350fpm.
We used to check the cabin rate of descent on the descent forecast page, if it was greater than 350fpm then we used to begin descent 15-20nm early to trigger the controller (about 1500' below CRZ ALT IIRC), once the cabin descent had begun then we could go back to VNAV PTH. The cabin descent rates of greater than 350fpm usually occurred as you described also where you are battling on into a strong headwind. One cure was to show TAI on for most of the descent!
A cross check is to check the cabin altitude to be lost vs time to go as part of your descent checks. So where your descent check says AIR COND & PRESS....SET, actually compare the cabin altitude to be lost with the PROG PAGE's ETA. If you have a 350fpm controller then it is about 3mins per 1000'cabin alt that needs to be lost. eg checks carried out at FL200, the cabin will be roughly 6000' at 3mins per 1000' to be lost means you want about 20mins to landing.
Sorry getting a bit carried away but to answer your question, the later 737's and the NG's with Digital controllers seem to be better. And the Airbus 320's are all completely automatic and never get caught out.
Standing by for Airbus V Boeing protagonists!