I just played a game where I initiated the 3rd repetition, and my opponent made their move before I could click the draw button, which removed my ability to call the draw. Realistically, I should have been able to call the game a draw until I made another move.

I'm sure it's not something that occurs very frequently, but that game resulted in a loss as opposed to a draw

That happened to me once.

@SlurpMcGurk
There is a useful settings, which I use myself too.
en.lichess.org/account/preferences/game-behavior
Here, the 5th setting is:
Claim draw on threefold repetition automatically
Never / Always / When time remaining < 30 seconds

#3 Good point.

Still, I think this feature would be useful.

This has happened to me!!

I don't like the automatic setting because i do not always want to claim a draw.

Maybe there could be a confirmation dialogue pop-up that asks if you want to claim a draw before it sends out the move?

Both USCF and FIDE rules suggest writing down the move and pausing the clock.

Online you can offer a draw, then play the 3fold move to claim a draw.