This is what happens in nature.
Two nearby masses will distort each other's shapes by their tidal gravity. If they are not tidally locked, then the gravitational pull on those tidally raised bulges produces a torque which acts to synchronize the spin rate with the orbital rate.
Think of it this way: if the planet spins faster than its moon orbits, then its rotation drags the nearest tidal bulge ahead, and therefore the moon's gravity tugs it backwards, which slows the planet's rotation. (This is what is happening with the Earth and Moon). The bulge on the Earth is also pulling the Moon forward accelerating it, which expands the orbit outward. Essentially this is just angular momentum conservation.
The strength of the tidal gravity follows an inverse cube law, while the timescale for tidal locking to occur is proportional to the distance to the 6th power. So it is a very
sensitive function of how far apart the objects are. This is why moons tend to be tidally locked, while planets generally aren't -- unless
they happen to be in a compact system like around a red dwarf star. Tidal locking around low mass stars is thought to be very common, and Space Engine reflects this. (Space Engine actually uses a formula to compute the tidal locking).
The eccentricity of the orbit doesn't change the rules for tidal locking very much, though for high eccentricities it may tend to produce a higher order of spin-orbit resonance. Whether the planet is more solid or liquid doesn't change the fundamental physics very much either, though it does change its efficiency (mainly through the dissipation parameter "Q" in the function given on wikipedia)
. The dissipation efficiency with the Earth-Moon system probably changes over geologic time, due to plate tectonics changing the arrangement of landmasses and therefore how the ocean tides "slosh" around the planet.
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Lastly, it seems to me planet systems that are billions of years old would change their motions slightly, without other orbiting bodies influencing them, never staying tidally locked forever.
Maybe, but then you must ask what mechanism would produce the torque to break tidal lock? Tidal locking itself is a stable equilibrium, since any change in the spin-orbit resonance would produce a counterbalancing torque trying to restore it, much like a spring pulled slightly from equilibrium. It's a very weak restorative force to be sure, but if you're talking over timescales over billions of years...
Things that can have some effect are massive earthquakes, which redistribute the mass of a planet and therefore the spin rate by conservation of momentum. But this is a very small effect. I might imagine on longer timescales that the movement of a mantle plume, or changes in ice coverage across the surface, could act to upset the spin of a planet. But again these are very small effects and tidal locking will beat them on billion year timescales.
There was a question that one of the planetary scientists at my uni was interested in investigating with some grad students, which was whether terrestrial planets in very close orbits (such that the rocky surface is vaporized on the day side and then freezes out again on the nightside) might involve such a large degree of mass exchange between the day and night side as to prevent tidal locking. But I'm not sure what the status of that work is.