If the band saw controller uses thyristors, whyis a bridge rectifier needed, except to power the electronics on the controller?
Why are 2 thyristors used, instead of 1 bigger thyristor? I hope they don't operate in parallel because I've heard any mismatch between them can lead to destruction of both. If you put in a bigger thyristor, be sure the gate current rating is about the same because thyristor come in regular and sensitive gate versions. It's not unusual for thyristors to be rated 600-1000 volts peak, just to withstand spikes from the motor. Also there may be a snubber network for those spikes, and it may have blown because they're sometimes made with a capacitor that can't stand the full voltage or a resistor that can't stand the power, and when a snubber blows out, other components may be damaged.
Unlike light dimmers, some motor speed controllers actually measure the motor RPM, either through a sensor or the motor back voltage.
If this is a Harbor Freight product, you may be able find the schematic at their website. And if it is something like a Harbor Freight product, it may be wise to replace the original power components with higher rated parts because some may be undersized. For example, my 1500W HF heat gun uses a diode to get the lower temperature setting (very common design), but the diode is rated for just 5 amps continuous, despite 8 amps flowing through it, and this product actually received UL approval. OTOH my ancient American made Sears Craftsman 6 amp battery charger (by Schumacher) uses a thyristor rated for at least 25 amps.