The Apple repair folks won't actually replace the trackpad itself, but they will adjust the tensioner underneath the trackpad. I had the same issue in my brand new Macbook Pro, and they were able to solve the issue.
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.
Best Answer
This is how the trackpad is designed. The trackpad is pivoted at the top, so it requires more force to press at the top than at the bottom, up to the point where you are pressing at the pivot which won't click the trackpad.