What I would do to check it is pull the module out of the firewall, leaving it connected, then adjust the climate control so it should be running the fan and use a multimeter to measure for drive current (actually it's a voltage measurement, between the pin and ground) on the base pin (pinout provided on the datasheet on that digikey link I gave, the casing itself seems to be the collector connection).
The datasheet isn't clear about which pins are
#1 and 2 on the first page but you can see they are offset at the end of the datasheet under the package dimensions diagram.
Edit:This brings up an issue that I hadn't thought about previously, that if they had electrically insulated the casing from the heatsink then you have to make sure that remains insulated, although from the pictures it looks like the rivets are carrying the collector current and so they need a good, clean contact to the PCB.
If there is no base voltage then something is wrong before the transistor. If there is base voltage, and collector voltage, but no emitter voltage, then the transistor has failed. If there is variable base voltage but the emitter voltage is constant, it also signals a different kind of failure, a short rather than open fault but that would leave the fan running but not the same range of variable speed.
If it's an open fault it also won't be getting nearly as hot if it heats up at all. To be clear, either open fault or shorted fault both mean the transistor is bad.
I don't have access to a picture of other components or a schematic of this module to advise about checking other components on it. Certainly if there are resistors they should be checked, and especially check for broken solder joints since this module does heat up quite a bit.
Bad ground could be checked by measuring resistance between motor ground and a good/clean chassis ground. It should be very low like less than an ohm but you may need to zero a meter to get rid of contact and meter lead resistance if you're worrying about exact value rather than only that it is low.
The main concern when soldering in a new one is that they seem to have used quite a large amount of solder on the PCB, so take care not to excessively overheat the transistor while soldering it in. Work fairly quickly soldering one lead, then letting it cool down before proceeding to the next lead.
I'd consider using stainless bolts, nuts and lock washers instead of what looks like a rivet to secure it so you don't have to drill it out every time it needs replaced if this happens again. Then again that raises cost while rivets are dirt cheap. Rivet or bolt it before soldering so there is no stress on the solder joints.
Computer heatsink grease should be fine.