As requested, WBsave, and buffy save (so you dont need to add barb techs) added respectively.
Yes I have seen numerous people bloat out that archery is seemingly pointless and you can just build 2 warriors, but that's a gamble and an archer guarantees safety. 2 warriors in a forest have something like a 15-20% chance to both die; if the archer wins at 30-50%, it'll always promote, and if it heals to 2.5 or over it'll have half decent odds to win again. Hell, there was this one game I had 3 warriors in a forest all die to 1 single archer and it was game over because of that. Similarly 3 warriors in a forest hill died to a spear that spawned at 2320bc and it was game over because of that.
in this map there's tons of open land with flood plains and all, the archers start spawning and circling your border at turn 11, im not sure how you would do that with just warriors; rush out into flat land and hope for the best?
I was even forced to defend in the city several times against archers and even spears later on; that's such a gamble with warriors because again you only have 50-70% chance to win, and if you lose they promote and have a shot to win again.
If you're willing to reload/replay the map for optimization then of course warriors are better. If you are going in 100% blind and don't know what to expect and this is your only shot and want to avoid reloads/replays; then archery seems pretty logical to me if the land looks like it can spawn a lot of barbs and be difficult to defend from.
When I play aggressive OR creative civs I change it up a bit. Aggressive civs have super warriors by building a barracks for only 25 hammers and then promoting to cover; this will give you substantially better odds and turn 40 into 65% and 70 into 85%. Creative allows your borders to rapidly expand and cover many many many more tiles early on from fogbusting, and not only that but give you much more warning when barbs are approaching; allowing you to react accordingly, meaning its also easier to do it with only warriors.
And of course, examine the situation. If you're relatively cramped or have close neighbours, and see several friendly archers orbiting your border, you have a good chance of being safe. However as you saw in my video that doesn't always apply as hatshepsut's archer walked right past a barb archer that bee lined my 2nd city.
I think 10-50% is a pretty reasonable choice, I sit somewhere around 25-30 nowadays. Quite often I tech archery not because of barbs but because of scary neighbors on my border that can plot at pleased.
EDIT: another note i'd like to add in is teching hunting + archery sure is a pain in the butt, that's a lot of beakers wasted and you should be looking at every possible solution to avoid that; build more fogbusters early before the settler, settle early horses/copper even if its a sub optimal location?
But if you start with hunting or have a reason to go early hunting (ivory, fur, deer, animal husbandry), then teching archery is significantly cheaper and I hardly think it's a bad play at all, even if you don't end up needing it, its a cheap tech. Especially once you start getting into the barb spear period if you haven't fully fogbusted yet.