To design an amp you need to start out with clear specifications of the amp's capabilities and parameters, such as RMS power, signal input level, supply voltage(s) available, load impedance, bandwidth, tolerable distortion level, .....
I'm sure that's workable, but I'm surprised you abandoned the original idea of using the two thermistors as a potential divider? That divider plus a trimpot could make a DC-energised bridge. A single opamp could detect bridge unbalance.
How small are your 'small batteries'? It takes a lot of current to heat a short length of nichrome wire to even a low red heat; maybe more than your batteries (which inevitably have internal resistance) can provide.
The battery is presumably a three-cells-in-series one and will need special charging/discharging control to avoid fire or explosion and to keep the cells balanced. So a dedicated charger/balancer will be required. The one in the link doesn't seem to have any call balancing ability, since there...
LTspice doesn't accept zero for the rise or fall time and will substitute its own times (0.1 sec in your case, which is what your graph shows). Replace zero with a practical value, e.g. 1u.