Impossible to claim draw by 3-fold repitition

I just played a 3 0 blitz game, and got a position where we repeated the same position FIVE times. This means there were 3 times I could have claimed a draw.

On the third, fourth and fifth repetition, I clicked "claim draw" immediately. Each time, it took a couple seconds to go through, and before the draw was officially claimed, my opponent made a nother move, canceling the draw claim. This seems like a pretty big problem, since as long as you spend less than 2-3 seconds per move, it is impossible for your opponent to claim a draw. Maybe this could be changed so that as long as one player claims the draw, even if the opponent makes another move, the draw still stands, since this seems like an issue on the site, not with my connection (I have about 150ms ping).

Under FIDE rules you are supposed to claim the draw then show the repetition move to the arbiter. I believe Lichess works the same way: offer a draw, then play a repetition to claim it.

That's odd, when positions have been repeated 3 times for me, there is an option to claim a draw, without the opponent having to click it as well for it to go through. Perhaps that was a bug? Are you sure the position was repeated exactly 3 times?

I tried it and it seems that if your opponent's move was the 3rd repetition, you are safe to draw

But if your move was the 3rd repetition, your opponent can cancel it by moving quickly

The 2nd scenario should be changed obviously as @Toadofsky points out the FIDE rule makes way more sense

As @Toadofsky suggested you can avoid the second scenario by offering a draw first and then making the move. Lichess will auto-claim for you in that case.