Your deck has under 40 cards, meaning Gardens are only worth 3 points to you. Your opponents have Provinces that are worth 6 each and they used a fast strategy (Council Room engine) to get them before you could end the game. If you want to rush Gardens and win early when they are worth less, you need a more effective way of depleting piles. Workshop is actually pretty bad at this on its own contrary to conventional wisdom, though it's OK in base only games. In a 3 player game, it's also somewhat unlikely you can manage to deplete all three piles unless somebody else is contesting you.
Your opponents also have the Council Room/Militia combo punishing the other players, which is pretty nasty.
Another problem is that you opened Militia. if you planned to rush Gardens in the traditional manner, it would have been better to open with Workshop and forgo Militia (their Militias would hurt each other for you anyway).
I think a better question to ask is "why go for Gardens here?" The engine seems too powerful and fast enough to either a) win regardless of the other guy getting Gardens or B) Contest the Gardens while also getting plenty of Provinces and winning on points easily. I do think a Gardens strategy works, but Council Room is pretty good vs a deck that plans to play 1 terminal and buy something cheap anyway.