How do four-week seasons work on MTGS, though?
Their four-week tournaments are all-alts. They've been doing this for over a decade, I think they've collectively abandoned vanilla. Obviously not the situation we're in.
Their first three rounds this season:
Modern NOEL: 4 card modern, deck must contain one card starting with each letter N, O, E, and L.
No untap: 3 cards (plus one basic land per turn can be played from nowhere), players skip untap step.
Backdraft: 5 cards, your opponent plays your deck; there are basic minimum decks the deck must be able to beat.
Their basic rules are a little different in a few ways too - most prominently, where we use "No deck that can win before opponent has completed first turn" they use "No deck that can win before opponent has begun second turn". And where we use "No deck that can cause opponent hand reduction of more than one in a single turn", they use "No deck that can cause opponent hand reduction before opponent has begun second turn". They also score all match draws as 2 points, even if they're 3-3 draws. Not that we need or should adopt any of this, just trying to paint the MTGS scene.
what does a new season actually signify?
It's a set of four weeks of alt matches, at the end of which the highest cumulative scorer (normalized to players in the round) is named champion and a new season begins.
I would suggest that banlists not mutate in a small way every round, but that a new banlist be defined at the start of each 'season' (with the right to define that as a prize for the champion(s) in the preceding season). The eventual steady state could be two vanillas, one upside alt, and one downside alt in a month - both vanillas having the exact same banlist (so the second v-round will be a strictly reactive meta to the first - the decks that had success in the first will likely not be viable because those decks will be the ones everyone's ready to cope with).