What map script? On a standard size continents this seems pretty wasteful. On Pangea or Oval its probably the best, but really Siam is a bit OP on those map scripts.
I agree tradition is the way to go because it benefits a lot from early food CS. I'd just build the monument. Sure it isn't that great if you meet culture CS, but at that point you have a rich man's problem. More culture always better.
I do this occasionally, especially if it can help you to fulfill a quest (such as a luxury or ending a civil war). You could try building a warrior instead of a pathfinder if the terrain is open, because you cannot gift pathfinders. He can probably meet those CS about as quickly, and get you an extra 12 turns of influence.
I'm on standard size continents++. The 2x pathfinder worked well in my current game- they finished mapping out my continent and met ~6 CS before I unlocked fishing. They then set out into the ocean in opposite directions, one of which quickly found the other continent and another 6 CSs. If I only had one pathfinder I probably wouldn't have found my initial 6CSs so quickly and before unlocking fishing. I also would have had a ~50% chance of not going the right direction to find the other continent, delaying finding the other CSs by maybe 20+ turns depending on how quickly I figured it out? Then again, occassionally the other continent is not accessible by connected coast tiles and my 2 pathfinders would be more wasteful in that instance.
I do like your idea of building a warrior first, though. They cost less production and can be gifted for the influence and would still be able to do some scouting. I did build and gift ~4 warriors this game to stay friendly with a culture and maritime CS and extend the culture/food yields until I was able to get quests completed and start building diplomats. Maybe something like this would work:
Warrior-shrine-monument-warrior
If you find a religious CS early you can delay the shrine and if you find a culture CS you can delay the monument, in both instances in favor of more warriors for gifting/scouting. Send the warriors out in whatever direction until they hit a dead end and then gift them rather than have them try to double back.
Overall, my current game is going well. I've nabbed Roman Forum and Summer Palace and got Hanging Garden as well just due to a well timed GE. I almost got Oracle but didn't need it, really. I'm closing in on Palace of Westminster and if I get it I think I'll be in solid shape. The only two problems I foresee are:
1) Austria is in the game and so will be a headache to fight over CSs with
2) I'm very small/tall at just 3 cities which means my unit cap is very low. Low unit cap + CS alliances means I'm a target and getting frequent DoWs (Indonesia, Austria, and Ottomans). My cities are very well placed for defense via terrain and nearby allied CSs so I'm in no danger of being overrun but the DoWs are wrecking havoc on my trade routes and ability to safely send out diplomats.