Assuming you're on the right wires, I agree.
Again, if you're indeed on the wires that go to the thermostat, then either there is a break in the wiring or the thermostat is bad.
Seems unlikely.
You have 24V at the two contacts at the furnace that go to the thermostat. Should be the red and white wires. When you connect them together and the furnace doesn't start, did you measure the output of the transformer? 24V there?
You could have a bad connection there in the furnace wiring. If a connection was high resistance, you could see 24V at the terminals, but when you connect them together, enough current would not flow to close the relay. Imagine the circuit being correctly working, but then a 10K resistor is put in series (simulating the bad connection) You'd still see 24V open circuit, but if you connect the terminals not enough current will flow to close the relay. Could also be some bad component too, eg the relay coil, presenting a higher resistance than should be there.
The other part, about the thermostat and/or wires going to it, no more ideas on what's going on there.