You raised the problem of metro stations being further than a walkable distance on either end of the commute.
If a subway or train service is at standing room only capacity right now even before you help new people to get to it then it's kind of a moot point anyway; you need more capacity before you can add more people.
Your first argument was that people couldn’t use public transport because of the last mile issue. That only affects a subset of passengers and is answered by folding bikes.
Your second argument is that it would impossible to operate a subway if passengers were allowed to bring a second cabin sized carry on bag on (that’s roughly how big a folded Brompton is).
Well there’s already plenty of examples where operators are already doing this successfully.
The London Underground for example operates just fine allowing folding bikes to be taken on at all times of the day.
They allow people to quickly reach mass transit hubs when none are near by, fold up and hop on. What’s missing is normalisation and support from transit providers.
I’d add, that it would be nice if Google/Apple would add multi-mode transit routing for maps so that you could tell when it’s quicker to cycle direct and when it’s quicker to cycle to a station and catch a train or bus.