I don't wait for tradition to build me the free monument every time. reason being: extra potential. The earlier you get into tradition and get to the growth tenet the faster you get off the ground. If I get a free culture ruin and open tradition at turn 5 then I find building a monument after my scout takes almost as long as waiting for the 2nd policy point to become available as 30/4 = 7.5 turns. Monument is usually 7 turns as well so I don't build monument and usually whip out a 2nd scout right away. However, if I don't find a culture ruin in about 8 turns or so I definitely build a monument. Reason being that it's not worth it to wait 25 turns to open tradition--that's a lot of wasted time I'm not getting that +3 culture. It's better to pay 1 gpt and build that monument. It's usually just a few turns anyway. Then later you get a free ampitheater which is far more expensive and automatically have a slot for the first writer.
if you build monument with tradition, you are actually losing potential, not gaining. As the free amphitheater is something that you rarely in a hurry with, so you can time it for the times where you don't have anything important to do, and it usually takes a turn or two. Its a big trade off, as you would gain a few extra turns faster legalism, but you will lose 5-6-7 turns worth of scouting - that means - less ruins, less gold from CS, probably less CS quests, worse area scouting (to see the best spots for expansion), and the more you wait, the harder it gets to get a free worker. And besides this - after you research pottery, you have other things to build, so I don't understand where you get the time to build after monument. Or you go only with 1 scout?
And your first slot for writer should be National Epic actually, so Amphitheater is something you can delay a lot even if you play a culture game ... Most of the time, they are the semi-useless thing that you need to build, because you want Hermitage and the Opera requires it.