I'm assuming you tried the same test while it's disassembled on the other two buttons and they work correctly, right?
That would pretty much point to the switch itself having given up after years of use. If you're comfortable with a little bit of soldering, it should be eminently possible to replace that broken switch.
A quick search didn't bring up any switches that specifically stated they were for your mouse, but it does appear those switches are fairly standardized. You'll have to verify the dimensions and position of the three pins on the bottom of the switch, but these from Amazon look like they may work.
Here's the link to that particular part, although I'm sure a more thorough internet search would probably turn up plenty more.
Amazon.com: RUNJRX Black Kailh GM 8.0 Mouse Switch for Gaming Mouse 4Pcs : Video Games
Good luck; let us know how it goes.
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