If it ran hot, part of that was probably the fact that the 6.3 V heaters ran at 900 mA, instead of the 300 mA. This is a full 3x as much power, just for the heater.
You could use it in a guitar amp, but I would probably just use a 12AU7 instead. The 5687 has a different pinout.
> It wouldn't be a good fit for the final output stage before the output transformer because of it's relatively low heft compared to a beam power tube or something, [...]
There are guitar amps out there with less powerful dual triodes used as output. Yes, they're low-power. There are amps out there using a 12AU7 for 500 mW of output power. It seems like such a small amount of power, but it can be loud--loud enough to damage your ears, loud enough to get evicted, not loud enough to keep up with a drummer.
I'm guessing this is because it was using early germanium transistors which would most likely be PNP. It was a long time before NPN transistors became the "default", because they were easier to make in silicon which came along later.
Tubes on computers from this era were used almost exclusively as amplifiers, inverting and otherwise. [1]
Inverting amplifiers are super helpful to turn the and/or gates you can build out of diode logic into nand/nor gates which then give you access to all of the other logic functions. But if you only need and/or for a particular circuit diode logic is much cheaper with much higher MTBF. [2]
Non-inverting amplifiers are useful to buffer signals when the required current draw is too high for a single tube/diode to push.
So in the LGP-30, you primarily see them being used as the inverters for the nor gates on it's flip flops, the amplifiers needed for the read/write heads of the drum, and inverters/buffers needed in the general logic. [1]
And FWIW, I don't really see why it wouldn't do a good job within a guitar amp. It wouldn't be a good fit for the final output stage before the output transformer because of it's relatively low heft compared to a beam power tube or something, but it'd be a good fit for an input stage before that. They really don't take that much plate voltage (the LGP-30 runs them with a B+ of 150v). Maybe they're simply too clean of an amplifier and guitar amps like that distortion of tubes that are more prone to voltage sag?
[0] If you want to follow along home, the "how is it using tubes?" portion of the schematics starts on page 11 here: http://www.bitsavers.org/pdf/royalPrecision/LGP-30/LGP-30_Sc...
[1] I want to add in two places using [1] that in addition to what I'm saying about amplifiers, diode tubes are also used as rectifiers in the power supply. There'll only be a handful of the in each design, but because they're used in the power supply of pretty much every design they're still ubiquitous even if they don't count for much versus all tube use in these computers.
[2] This use of germanium diodes in logic came pretty late in tube computers though, really in the late 50s at the cusp of transistorized machines. Royal Precision's next computer, the RCP-4000 was basically the same design as the LGP-30 but with tubes swapped out for transistors. Leads to some goofy schematic choices like the negative voltages being at the top of the schematic pages because it lines up better when compared with a tube based design.