I have a Glacier Bay Niagara Flapperless Toilet. The flush handle (part # C2228-3) keeps falling off the shaft. How can I get it to stay on?

For a half flush, push down handle and release. For a full flush, push down handle and hold then release.

Yes you can access the hinges, the seat bolts are under cover on hinges. Refer to installation instructions that came with toilet. page 8. Don't have the paperwork go on line to Home Depot on line and look for your toilet. the instructions are there.

ensure that the supply line is clear. shut angle stop valve off and remove line from bottom of tank place line into a pail and open angle stop valve, if clear water will flow quite readily. if not replace line. the fill valve is available at home depot. personally I would replace toilet with a quality American Standard. if your toilet is not a Glacier Bay good luck finding repair parts.

https://www.homedepot.com/p/Fluidmaster-Flapperless-Toilet-Fill-Valve-for-Glacier-Bay-Toilets-703AP4/202510398

Glacier Bay Toilets are made in China by "various manufacturers" a local plumbing parts supplier is your best bet.

There should be a adjustment screw at the top of the fill valve, screw it up or down to adjust the tank level.

check water level and length of puller

You should be able to find this at a specialty shop if not lowes or Home Depot. Google your part online and you should find it.

Does the tank lid have a seal on it to keep water from coming out?

It is more than likely the flapper. Easy fix. Ck the chain for too much slack. Good luck_Ned-