In theory tying the 2 wires at thermostat together, red and one of the other wires together, generally white, bypassing tstat, if t-stat dead, would cause heater to come on. However unfamiliar with 3rd wire, or why, pilot light-AC signal?
My hesitation is Ive read on some t-stats jumping while connected can blow them, I assumed on electronic tstats but yours & 3rd wire? Without seeing it-tracing out path unsure. I wouldnt want to tell you to do something that might cause magic smoke.
No reason to and I wouldn't cut wires at t-stat- there screwed on? but unsure which is the return wire. As there are only 2 terminals, R & W, I assume the black & white wires are tied together at the W or bottom screw. At heater tstat wires are the 2 blue ones, you have blue++++ to red +pos to tstat. The other blue wire has a yellow wire? hard to trace in picture, Im not going to say the white wire at tstat is same yellow wire connected to blue wire at heater. Confused why a 3rd wire? Its a 2 wire thermostat. Thermostat simply makes a connection between the 2 wire when temp reaches setting. testing should take all of 5 minuets
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I am assuming your test light is polarity correct and red wire is hot.
TEST
Meter would be better but...Since you have a test light, place its clip on another ground source, not on tstat.
-Retest you still show power at red wire- small screw top left marked R terminal,
-then turn on tstat-test for power at other small bottom left W terminal (black/white wires).
If power shows up at W terminal tstat is ok. Problem in heater or wiring to it.
If power does NOT show up at W terminal continue, problem likly the tstat. To verify tstat is issue
-Loosen the small screw at the top left terminal marked R. Remove the red wire.
-Loosen the small screw at the bottom left terminal marked W and remove the wire(s).
Hold the ends of all wires together, It may take up to a minute for the heater to come on. If heater comes on the t-stat is at fault.
Simple inexpensive replacement fix. Done.
TEST2
If heater still does not come on with wires at tstat manually tied together (or tstat tests ok). As you have power going to tstat, another simple test before assuming fault in heater or other issue is to test return wire from tstat to heater isn't broken or disconnected somewhere.
-disconnect the wires from each other at tstat if still connected
-Cut the crimped on caps off the blue wires at heater. (They can be reassembled with simple twist on wire nuts later). Test which blue wire is feed/hot to tstat, twist back together (I assume blue/red is hot)
One blue wire will always be hot, the other only when tstat closes.
-Test the other blue wire connection (I assume blue/yellow wire.) I should not be hot. Test the yellow (why is it yellow?), should be no power, reconnect wires togther at tstat (or turn on tstst if ok), power should show at yellow wire. If it does NOT show power and wires at tstat tied together you have a break in return wire.
To verify
TEST3
Disconnet red/yellow wires from blue wires. Hold the 2 blue wires together-heater should start within a minute. If it does not come on problem is heater.
If it does come on then its the tstat return wire. Though you might retest tstat.
Depending on what test revel-If tstat is ok and/or tying wires together you never see power at the other yellow/blue wire AND touching the 2 blue wires together fires heater I would simply run a long piece of wire connected to the W terminal of tstat and connect other end to other blue wire in lieu of yellow wire. If it works means either finding break in return or rerouting a new return wire.
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