I've just played about 10 games on this kingdom trying to get the golem/counting house strategy to work.
The best I've been able to 4-province is around turn 18, and that happened with some crazy luck. Turn 3 $5 for a quick counting house, then drawing silver/potion/copper/copper estate turn 4 (i.e. EE as my 11th/12th cards) for a golem turn 4. That's as ideal as it gets, and I still only 4-province'd on turn 18 in that game.
My priorities were then to get 1 more counting house on $5 and more golems on <$7P, otherwise Province if possible, or silver/copper. I'm still not sure if you should buy silver or copper with $4 on early turns.
Still, even with that amazing start, I had numerous hands where I'd reshuffle and draw Counting House + 4 copper. Or on the 2nd reshuffle, draw counting house + copper, returning maybe 2 copper (from the previous P 2c, 2 E hand) and gaining $6 only.
I dunno, it seems super inconsistent to me. One game, I had my 4th province on turn 25, simply because the shuffler hated me.
Counting house/golem is not a gameplan I'd advocate on this board. I'm not sure what would have to be on the board to advocate it. Maybe Goons, or an opponent's mountebank? lol).