Why does my INDX only drop tools at the dock sometimes?

That wiggle is your problem. I went through the exact same head scratcher a few months back, flaky releases that got worse over weeks, no pattern to it, worked fine then wouldn't let go twice in a row. Pulled the cover, found the same thing you did: the little wire that hooks into that lever had chewed away the plastic it snaps against, so the lever could move a bit without actually shifting the gear behind it. The dock magnet pulls the lever, but with that slack in there the movement sometimes falls short of what the mechanism needs to let go of the tool.

The reason it seems random is that the slack sits right at the edge of working. Vibration, how the head lands on the dock, even which direction the wire gets tugged can be the difference between a clean release and a half release. And don't worry about it happening on both sides of the dock, that's expected, since the lever lives in the toolhead, not the dock. If it only failed on one side I'd be looking at the dock itself. Easy way to confirm before you commit to anything: with the cover off, push the lever by hand and watch whether the retention mechanism moves all the way through its travel. If there's dead travel at the start of the lever's motion before anything else moves, that's your slack.

For the fix, I get not trusting your hands with glue in there. Mine was simpler than the microscope job it sounds like. I gently bent the little hook on the wire so it seated tighter against the lever, taking up most of the play mechanically, then put a single tiny dab of thin glue on the joint as insurance. Thin glue wicks into the joint on its own, so you don't need to be a surgeon about placement. Just keep it away from the pivot and let the lever swing freely before it sets.

It's a cheap part if you'd rather swap the whole assembly instead, and it's only a few screws. Mine's been solid since.

Same story here, the wire and lever was exactly what was wrong with mine. One thing I'd add: don't wait until the wiggle is obvious. I checked mine around 5k changes just out of curiosity after fixing it and there was already a hair of play starting, way before it would've caused a failure. Now I peek at it every few thousand changes and it takes two minutes. Also if you do the glue route, cycle the lever a bunch of times while it's curing so it doesn't stiffen up on you.

On mine the wear was on the lever side, the hole the wire sits in had gone oval shaped, so bending the wire was never going to take all the slop out. Just ordered the replacement assembly instead of fussing with glue. Took me maybe twenty minutes with the cover off, and the flaky releases stopped that same day. While I had it apart I also cleaned the grit out of the gear teeth with a soft brush, there was a surprising amount of filament dust packed in there.

Mine did the same slow creep in failure rate, and the thing that finally convinced me it was mechanical wear rather than anything electronic was that the half releases got more frequent on a smooth curve over weeks, which is exactly what you would expect from plastic slowly rounding out. One tip for after the fix: keep a running tally of changes for a few days and jot down any partial ones, because if the count starts climbing again you know the slop is coming back and you can catch it before it strands a tool mid print.

Chiming in because mine came back after a fix. Swapped the lever assembly, all good for about a month, then the flaky releases started creeping in again. Turned out the magnet arm on the dock side had loosened off a hair, so the pull wasn't as strong as it should've been. A quick snug of that screw and it's been fine since. So if yours acts up again down the road, don't assume the new part failed, check the dock side too. Two minute job.

Update: pulled the cover again and looked closer at that lever. The wire itself seems fine but the hole it sits in has a slight oval shape on one side, which sounds like exactly what the third reply described. Guessing bending the wire would just be a temporary patch then. Ordering the replacement assembly and I'll report back once it's in.