Yes, the resistor is a shunt to ground so it acts like a filament light bulb, just enough to trick the flasher relay into thinking it has real light bulbs out there instead of LEDs. The flasher (either mechanical or electronic) is using the light circuit as a path to ground and expecting to see enough current flowing to pull in the relay. When you install LEDs they draw far less current and/or provide less of a path to ground, so the ground signal is less, meaning that the flasher relay doesn’t have enough strength to pull in properly - THAT is why the passenger side, with longer wires, didn’t work right.
The resistors are only used on the directional circuits - they are used only intermittently (flashing) so they only get gently warm to the touch. If you were to use, say, 20watt resistors with heat sinks they shouldn’t get warm at all. Mine are screwed into a 2”X2” frame member as a heat sink so they don’t get hot. Autozone sells them just for this purpose in the light bulb aisle.