I have also thought about thread like that, either a mod or even an official oprion.
"Classical mode" may be too much for the devs to put in the game of their vision, allowing to entirely avoid it. But what if we simply had the option to retain the old culture into the next age
by handling crisis really well?
I mean let's see
1) It makes sense within the game's narrative: some exceptional civs may survive dramatic era transitions after all
2) It makes historical sense - you can totally find many examples of multi-era civilisations, not matter how you define both terms. Maya and Khmer (already in civ7), Japan (it even has the same dynasty for 1500 years), Vietnam, Armenia, Georgia, Jews, Assyrians, Basque, Ethiopia, Ireland, France and many European cultures etc (not to mention Persia and China in a way IMO, though that's controversial)
3) It provides an additional incentive to handle the crisis well
4) It makes retaining old civ feel "earned" and actually "standing the test of time"
5) AIs would sometimes do that, but you could never predict who and when, so when you enter new era it is even more unpredictable adventure
6) It would greatly simplify players' and Firaxis headache of seeking historical civ transitions, filling TSL maps etc.
The main problem with the civ switching imo is not "I can't unlock preferred civ immediately" but "I am forced to abandon it". The former is not contrary to the basic game design and psychology (make player earn something) whereas the latter is (forcefully take away toys from the player). The former is also far easier to justify historically and narratively than the latter. Sure, I'm fine with having to play as Slavs before I become Poland, in fact I think Poles would find this very cool, but forcing them to switch to Russia or Germany in the modern era is the worst insult civ game could make - you basically confirm the Prussian propaganda trope that Poles can't govern themselves and their failure against imperial colonialism is inevitable