#1 Failboat: 68 points (4 Provinces, 14 ▼, 4 Dukes, 3 Duchies, and 9 Estates); 38 turns
opening: Silver / Monument
[46 cards] 4 Dukes, 3 Monuments, 1 Philosopher's Stone, 1 Rabble, 7 Coppers, 7 Silvers, 2 Potions, 5 Golds, 9 Estates, 3 Duchies, 4 Provinces
#2 zahlman: 58 points (4 Dukes, 3 Provinces, 5 Duchies, and 5 Estates); 37 turns
opening: Silver / Militia
[44 cards] 5 Militias, 4 Dukes, 2 Philosopher's Stones, 1 Expand, 1 Monument, 1 Rabble, 10 Coppers, 6 Silvers, 1 Potion, 5 Estates, 5 Duchies, 3 Provinces
First I'm going to analyse this based solely on the summary.
It seems unfair to characterise this as Duchy/Duke/Militia vs. Monument/BM. Your opponent is only off 10 points in the Duchy/Dukes, and they more than cover that with Monument, as well as picking up an extra Province.
You have far less money than your opponent: no Gold to your opponent's 5! This game is always going to run a long time, and the lack of sustained buying power will really hurt you. The length of the game will also let Monument pick up a lot of points.
Militia isn't such a scary Attack here, as with so many Victory cards in the decks there will quite often be something to discard. You also have a lot of copies, and I'd have expected you to have some problems with collisions.
Now I'll take a look at the log. You buy Duchies very early. Turns 3, 6 and 7 are Duchies, after which you have a deck of 7 Coppers, 3 Estates, 3 Duchies, 2 Militias and 2 Silvers. You pass turns 8 and 10. Your opponent has 7 Coppers, 3 Estates, 5 Silvers, a Gold and a Monument. By buying Treasure your opponent keeps their options open: when you go so hard into the Duchies, you're declaring that the game will last a long time, so they can plan accordingly, picking up more Monuments and aiming to buy Provinces whilst holding you to a small Duke lead.
Late game, it's not clear what the Expand is doing for you, as you don't have any good targets: there are no Duchies left, and you only have a single Rabble that you might possible want to turn into a Province.
Philosopher's Stone is always fun.
It seems you buy some of the Militias quite late, and only have one collision, so terminal numbers are probably fine. I'd say that the main thing is misjudging the length of the game, and leaving your economy in such a state that it's hard for you to compete later on. I probably like Monument more than Militia here; that's probably something the simulator could weigh in on.