MacBook Pro Left click not working on trackpad as well as mouse

The solution was to disable the internal trackpad. The trackpad, one of the older style pads with the separate button along the bottom edge, had a malfunctioning button and that seemed to be at the heart of the problem.

As soon as I disabled the trackpad the mouse would word. And as soon as I re-enabled the trackpad the mouse would cease to work.

For details on how to disable the trackpad see this question and answer.

Before you do anything hastily, try resetting your NVRAM: https://support.apple.com/en-us/HT204063

This will clear several hardware-related settings and may resolve your situation.

If not, try creating a bootable installer (https://support.apple.com/en-us/HT201372), boot from it and check if your trackpad works there. If it does, backup and reinstall your OS.

The alternative would be to open up the device and check if the trackpad connection is properly attached. Sometimes (although very rarely) the plug can become loose, likely due to radical temperature changes. You should only attempt this if you're comfortable with electronics repair and following a guide on a site like iFixIt. They also have parts for self-repair.

If the latter doesn't help, sometimes disconnecting the battery and reconnecting it works, as this will clear any battery-buffered settings remaining.

In case all else fails, I suggest purchasing a replacement part through iFixIt and following a guide to exchange the trackpad – if you're at all open to self-repair. Having it exchanged by a shop is always going by to be vastly more expensive. The upside being that they're usually required to provide a certain warranty period for repairs. This is not a complicated repair, so you may want to try it yourself.