Thank you to everyone for the helpful tips. I was able to get my spice simulation working. Turns out I had modified some component values and forgot to set them back.
Reducing the value of R4 got the circuit running. However, it still does not replicate the physical circuit very accurately. I guess this is not surprising since I don't have accurate models for the LEDs or transistors. It was good enough to get an idea of what was going on though.
I changed some of the capacitor values to reduce the frequency to something sensible.
I never had an issue with the frequency being off in either the physical or simulated circuit. The duty cycle was very low but the frequency was around 2Hz. It looks like in your circuit the values for the emitter and collector resistors are swapped with the values for the base resistors. That's probably why the frequency was off.
I was able to increase the value of the parallel resistance across the LEDs to 4.7K which makes the circuit work. My understanding is this allows enough current to flow to make the circuit function since the LEDs are basically an open circuit when turned off.
I then had to decrease the value of R4 as suggested as the pot was at the bottom limit of its range. Tacking another 10K across R4 gives me a good range with the pot.
The duty cycle was still very low. Adding 22uF across C2 gives very close to 50% duty cycle.
I would like to get the indicator on the tachometer working if possible. It will stay an incandescent bulb since there are no LED replacements in that size, and it needs to be non-polarized based on how the bike is wired. I've attached a diagram of how I think I can make the indicator work. My idea is to use a transistor to short out the unused signal when the other one is active to give the indicator a path to ground. For example, when the left signal is active, the top MOSFET will be on, shorting the right signal to ground and allowing the indicator to light without letting the right signals light.
(I realize my usage of the words "indicator" and "signal" might be confusing to those from the UK. When I say "signal" I'm referring to the actual directionals that other motorists see, and when I say "indicator" I'm referring to the little light on the tachometer that lets me know the directionals are active. Sorry about that)
Does anyone see any issues with this solution?