Why does my AC blow warm air when there's water sitting in the drain pan?

Yep, your read on the float switch is right. It breaks the call for cooling when water hits it, so the condenser outside stays off while the indoor blower keeps running. And it doesn't take much water at all. A shallow puddle in the pan is plenty to trip one, so don't judge the clog by how big the blob was that came out. That slimy little thing you pulled out might have actually been the clog itself, or at least part of it. That gunk is basically algae sludge and it comes out in chunks like that.

If the line still isn't draining well, go back at it with the vac. The trick is getting a good seal where the hose meets the drain pipe, so wrap a rag around the connection or use some tape to keep air from sneaking in. Run it longer than a minute too, sometimes it takes a few minutes of steady pull before you hear water and gunk rushing through the hose. If the outside end of the line points down or has a trap in it, that can hold water and mess with your suction, but keep pulling anyway. A wet/dry vac can usually drag a clog backwards through the line even if it's up near the air handler, since the line runs downhill toward that outlet.

One thing to keep in mind though, float switches do fail on their own sometimes. So if you get the pan dry and the outdoor unit still won't come back, the switch itself might be the problem, not the drain. Easy to test by bypassing it for a minute to see if the condenser kicks on, though you don't want to leave it bypassed with water sitting in that pan because the secondary drain will eventually overflow into your ceiling. As for the tablets, drop them in the pan itself near where the water pools. They dissolve slowly and keep the algae from building up in the first place. If there's a cleanout opening or standpipe on the drain near the air handler, you can flush some water down that too after the tablets are in, just to push everything through.

I'd push back on the part about just keep pulling from the outside end no matter what. On my old system the drain had a couple of elbows and a long horizontal run before it got near the air handler, and suction from outside was basically useless, I pulled for ten minutes and got nothing. When I cracked the little cap on the tee right by the unit and vacuumed from there, the clog came out in seconds. If the standpipe or cleanout is right there next to the switch, try the vac from that end first instead of fighting the whole line length from outdoors.

Something similar happened to us last summer and the weird part was the big copper pipe going into the box in the attic had frost all over it, like a frozen bottle. Turns out when the cooling part quits but the fan keeps pushing air, the coil can ice up, and when it melts you get way more water than that little pan was made for. So if you see ice on those pipes, let it melt fully before you test anything. How do you tell if the pan water is actually draining versus just sitting there soaking in slow?