yup, after your opponent went potion you could have just rushed minions and destroyed him. (open woodcutter/silver). MAYBE woodcutter/lookout.
This board is a perfect opportunity for you to learn how to evaluate.
First, there are 2 incredibly powerful self combos, alchemist and minion. (hunting party and governor also fall into this category). In most games, losing the split on one of those means losing the game. In the same game, because of the /discard your hand attack of minion, it is clearly superior to alch. Even if he gets all 10 alchemists. Play 2 rabbles, then a minion, and bam, he's back to square 1.
Second, there are some minion powerup cards. lookout and upgrade to thin your deck, and +action (bazaar) with +$ (woodcutter) and throneroom. meaning you can get extra money and buys in the middle of your minion chain. This makes minion clearly the most powerful strategy.
third, There is brutal attack engine potential with bazaar and rabble/ghost ship. Ghost ship is the nastiest attack in the game. Any deck that plays a ghost ship every turn will usually win. Minion is a decent counter to ghost ship however. And ghostship/rabble doesn't work great because he can just put green or copper on top of his deck that he doesn't mind you rabbling. (plus early on he can put estates on top to make sure lookout hits them.)
4th - Woodcutter is the only + buy here. In a game like this where you may notice nobody bought a terminal before turn 14, you want to open woodcutter for sure. I suppose you could skip it because of upgrade, and just upgrade estate into 1 later (as your opp did). But in most minion or alchemist games (especially with villages or thronerooms) get those terminal silvers early so you dont have to waste money later. Many minion games come down to a 4/4 split. And if you throw in a woodcutter and steal the last 2 minions on a 10$ turn, you win. And you wont want to waste a 5$ buy on an upgrade in a minion race. (until after you win).