The first question you need to ask yourself is the following: does the spacecraft need to know its own position? Many, if not most, spacecraft do not need that information. The ground team simply needs to know when to schedule maneuvers, which means that only the ground team needs to know the position of the spacecraft, not the spacecraft itself.
If the spacecraft does need to know its own position, then a propagator in itself is not sufficient for onboard orbital determination (OD). In the subject of orbital determination, as defined in "Statistical orbit determination" by Schutz et al. 2004, the solution to the problem includes not only the position and velocity of a spacecraft, but also the uncertainty of that position and velocity. An example of how this might work is available here: you'll note that the plots include the error between the true spacecraft state and the estimated state (green dots), along with the uncertainty (red line). If using any kind of propagator, may this be the SPG4 which was initially release in 1988, or the latest and greatest models, the "solution" will only be the position and velocity of the spacecraft. As per the above definition, that does not correspond to a fully defined solution of orbital determination. Instead, it corresponds to a "propagated spacecraft state", which may be literally hundreds of kilometers off compared to the truth. For example, the difference between a non-J2 effect and a J2 effect is 0.097 km in just one day. I encourage you to run a few different simulations in NASA's GMAT to compare the final states of spacecraft using different fidelities of the harmonics and of the drag.
More specifically, an SPG4 propagator may give the spacecraft some very rough estimate of where it is, but the real-world spacecraft dynamics are far too complicated to be determined by any propagator by itself without any measurements, and without running an OD filter.
It would need some measurements of the world around it (through GPS readings or ground tracking passes) in order to accurately determine its state, and the uncertainty in its state. Moreover, if a GPS module is considered power hungry, then the computation needed for an SGP4 would also likely be considered too power hungry.
LEO birds may be equipped with a GPS module which can handle the velocities of spacecraft (they are usually ITAR restricted but nothing prevents you from designing your own chip). If equipped with a GPS module, two strategies for OD are possible.
First, the spacecraft could store onboard each of the measurements and downlink them to the ground on request. The ground team would then determine an OD solution independently in order to accurately determine the orbit of the spacecraft. The team will then upload a maneuver file based on ground propagation of the spacecraft's trajectory.
Secondly, if spacecraft needs to know its own position without ground support, then it needs to run an onboard Kalman filter. The Kalman filter will allow the spacecraft to infuse GPS measurements with the expected model of its dynamics (i.e. Earth gravity field, drag model, position of the planets, etc.), and compute an estimate of its position and velocity (and optionally other parameters), along with an uncertainty of its state.
When doing spacecraft navigation in operations, an OD analyst will run many different filters with slight variations in the dynamical model. For example, in the case of the GRAIL mission around the Moon, analysts simulated small maneuvers in their "truth model" in order to account for errors in the gravity field which weren't fully understood by scientists and engineers during the mission.
