without actually using the meter, it's difficult to say but I will say this..
IF the volatge was measured without a load, the reading may be incorrect because solid state will allow voltage through but no current. The way to test the 24v in solid state is to have a known load such as a relay. Measure the voltgage with the relay.
If the measurement was made while the furnace was locked out, the current to the R terminal may have been interupted by the control.. This is how it stops the unit from running.
IF I were there, I would test the circuit by removing the thermostat
and thermostat wiring from the equasion and simply jump R-W. If it fired, I would repeat the process several times. If there is a safety locking it out such as flame failure, I could go from there to determine if the flame actually came on, and then back off or could begin troublshooting from there.
My bet, is it would work with a jumper, the flame would ignite and go back out telling me that the flame sensor simply needs cleaned.
Now, I said all that because I dont know what his steps were. I am not saying he is wrong because there is no way from here to tell, but 13v sounds like an unloaded solid state type reading.