This sounds similar to an issue i'm having with my 2000 T4R with the manual AC unit with the 4 speed blower fan switch slider. I had to wiggle it to stay on, but now doesn't turn on at all. I replaced the blower resistor thinking that was the issue, but the issue remains.
I took off the center console and looks like the piece the lever is attached to is broken, the lever is kinda drooping. If i push up on it with my finger, it tuns on. The part looks similar to the picture you posted above. Not sure if I can just replaced that part or if i need to replace the entire AC unit. Found tons of used units on eBay but since these manual units have cables to the recirc vents and another cable for the hot/cold knob that pulls the cable that opens/closes the heater control valve. I've yet to find a DIY on replacing these manual AC units
Here's a video of the issue:
2000 3rd Gen 4runner manual climate control - YouTube