There is no easy way to answer your question from a distance with precision. It is largely dependent upon how hot the 7812 gets (obviously) which is a function of several things. The idea behind the whole heatsinking notion is to get heat out of the regulator package. This is or can be a challenge because the mounting/dissipating surface is only about 1/2" x 1/2".
1: How much current is being drawn by the circuit being supplied the +12VDC? These regulators (there are newer, more efficient ones) operate in a way where they have to "throw away" at least 2-3 volts from the input supply. You give them (for example) 15 volts, they output 12 volts, the rest has to go away as heat. If you pull 1 amp, then the package has to dissipate volts * amps = 3 watts. No getting around it. If you give them 18 volts, they need to dump 6 watts as heat. That may not sound like a lot of power but the question is always "over how big an area"? I think they can be supplied up to 35 volts as their max rated input volts. Then the package would have to dissipate 23 volts * 1 amp = 23 watts. That would be like a Deluxe Reverb in 1/2" x 1/2". HOTTT.
The most practical approach is to look at what is being supplied the 12 volts. If it is a few op-amps, I would stop your worrying right now and use the silpad. Those use tiny tiny current. Are there LEDs? 2 of them or ten of them? Just like we do when adding up heater currents of tubes to see if a particular transformer with a particular amp rating on the heater winding can light up the tubes....we just add up the requirements and get a feel for whether we are exceeding specs.
2: What's the ambient temperature? Inside an amp chassis, things are probably hotter than they are in a normal room. That makes the heatsink's job harder, because whatever type of heatsink you have works on taking "hot" to "ambient" and can never do better.
3: Let's also not forget that line voltages tend to be higher, which goes back to raising the input voltage level, while the output is still supposed to give only 12 volts. This increases the heat load on the device and thus the heatsink.
Forget about the reliability of the device itself, these things even in commercial grade are phenomenally reliable.
The silpads are pretty good. Personally, I have never quite trusted them as much as the goop + mica insulator but I am old school.
If it were me: The regulator (and if present, its negative voltage brother, a 7912) is supplied DC from the rectifier. If I wanted to reduce the heat load on the thing, I would install series diodes in between the rectifier and the reg input. Either a string of ordinary diodes or a 3-5 volt zener in between the rectifier and regulator input. Every ordinary silicon diode would drop about .65 volt from the input. Of course the great thing about diodes is that they are very cheap. I'd work to knock 2-5 volts from the regulator input with 3-7 diodes and call it a day. Your only consideration is mechanically mounting them but that's not a big deal. Diodes like this mount on their own leads all day long.