Would the meta eventually reach equilibrium in terms of the proportion of what decks are used?

If they stopped developing people would stop playing. I know I only started playing again because of tgt

I'd go with no. Hearthstone is a game where even decks that revolve around a very specific strategy aren’t using even half of their deck towards that strategy, decks like Secret Paladin and Dragon Priest. Most decks are much less focused than even that, just goodstuffs.dek. That's very strange for a card game, usually the numbers are more like 90 percent of a deck working towards a strategy. So in Hearthstone, as long as there's room for so much filler, there will always be infinitely different ways to find an effective side strategy with the possible synergies in all of that goodstuff. And at this point, the decks with strategies involving decks that can spit out their hand and have their cards boost eachother all basically suck, mechs and murlocs and whatnot.

Just going to give examples

* Meta turns into nothing but Secret Paladins
* Some new deck is created that absolutely ruins them
* The meta shifts - Secret Paladins lessen, other decks become played due to Secret Paladins not being around as much
Rinse-Wash-Repeat

Eventually people are going to stop finding new decks.

Compare this to music. Yes Hearthstone has more limits but constant meta shifts would bring new decks for a good bit.

I could only imagine how boring arena would get after a year or so.

You say develop I say deteriorate.
Only thing exciting about this game is the 1 month chaotic meta when new expansions come out, and the 1-2 games of Tavern Brawl each week before you get bored of it.