You have to level up the shoot arrow ability once before you can shoot arrows. When you start as an archer you can only melee until you gain your first level, the same as the other 5 classes. This is, admittedly, not particularly intuitive.

No, oops, I didn't communicate effectively: I leveled up and got the arrow skill. The skill itself didn't work.

https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=878660122

Here's a screenshot of what I'm working with. Pressing 2 doesn't work, enter doesn't work, clicking on the enemy doesn't work, etc.

The problem is the wait icon is missing without the touch controls (Compass), so it doesn't change to the arrow icon as it isn't visible. You need to press 2, followed by the space bar.

Hit <space> to fire the arrow. Here's my attempt at shooting an arrow for the first time. (The link should be to the time I upgraded the archer and then try to shoot shortly thereafter.)

Here is the video from the beginning:
https://www.youtube.com/watch?v=OMLV3r_L_PQ