Would a bad limit switch cut power to the thermostat?

It's possible depending on how your furnace controls are wired. Sometimes the limit switch will cut the 120V power supply to the 24V transformer that powers the thermostat, sometimes it's wired to cut the 24V power from the transformer to the thermostat, but you'd need to look at the schematic for your furnace and verify that the actual wiring follows the schematic. It's possible that it's been modified.

If the limit switch was tested bad, not just guessed bad, you are going to have to bite the bullet and replace it. It may be easier to test the limit switch than to trace out all the controls.