Use a 2-wire line voltage thermostat on a 24v / low voltage 2-wire furnace
Plain old mechanical thermostats don't care, but...

They are simple switches, whether they are line voltage, millivolt, or 24V.  However, obviously, they have different current ratings; I wouldn't lose any sleep over mixing up millivolt and 24V, but those must never carry line voltage.  

However, your thermostat with the single dial and the "large, black and white numbers" is not a plain old mechanical thermostat.  You are talking about some sort of smart thermostat there.  If this thing is battery powered, that's a possibility; but more likely it relies on a "C" wire or on current leakage through a large resistance heater (which would be pretty easy to do for a line voltage 'stat). These won't work on the wrong voltage.  

Why are you bothering? There's a huge variety of 24V thermostats

I gather you spotted one you like, but if you do some research I bet you can turn up a comparable one that runs on 24V.  Probably ten of them.   Some that use batteries, some that use C-wire power, and others that allow leakage power through a clunky old 24V solenoid like you'll have on an old furnace. 

As far as price, line voltage doesn't buy you anything.  There are very inexpensive thermostats available for all types.