What furnace and model number?
Is circulating pump running?
Is system on zone valves?
Is boiler cold?
This can be as simple as following the wires. The power for the thermostat originate somewhere. Track back to where this power starts.
Do the zone valves have little levers or dial on the side or end for manual open?
The boiler control power can be itting at the valve waiting to make contact through a end switch when valve opened. The power for the stats can simply work the zone valves, and the fact the valve opens turns on the boiler.
Now we know transformer on aquastat is good, transformer for zone valves good, the zone valves are open, the stats work etc.
Now is there a vent damper on the vent pipe that opens everytime boiler fires?
Back to aquastat, Terminals R and B in the center. That is a normally closed switch . We now check B to ground for 24v , R to ground for 24v, and Rto B for 24 (if open)
Ok last test use a short wire or jumper and briefly touch T to TV on the aquastat. See if relay clicks to bring on circulator pump
Bad K1 relay in aquastat. 24v available and sent direct by jumping yet circulator relay doesnt pull in.
Looks like you need another aquastat