You definitely have the right idea and understand some of the various ways to propagate orbits:
- Keplerian orbits
- TLEs with SGP4
- numerical state vector propagation
But all of these turn out to be approximations.
Keplerian orbits are conic sections and assume two bodies (point masses or spherically symmetric mass distributions) and nothing else. An elliptical orbit will execute over and over without any changes.
Propagating TLEs with SGP4 is way more accurate because SGP4 contains a pretty good model of the main bumps and wiggles of Earth's gravity field, and a pretty good approximation of aerodynamic drag. If you are in a high orbit (period greater than 225 minutes) then the "deep space correction (SDP4)" will also implement a pretty good approximation of the average effects of the Sun and Moon's gravity on the orbit as well. For more on that see the (currently unanswered) question How do “Deep space” corrections in SGP4 account for the Sun's and Moon's gravity?
However all of the pretty good approximations are just that, and no more. So while SGP4 can spit out state vectors with a dozen decimal places, none of those are "correct" so starting with one of them and propagating with a fancy orbit propagator will always have some significant amount of error.
"fancy" numerical state vector propagation is what is done when the best possible accuracy is required. It will have a finely detailed model of Earth's gravity field, up-to-the-minute space weather predictions to model Earth's atmosphere (and the spacecraft's aerodynamic characteristics) for a more detailed estimation of drag, and have the exact positions of the Sun and Moon and calculate their gravitational influences in detail as well.
But "garbage-in, garbage-out" still applies. If your initial state vector is of limited accuracy, then integrating from there will not be more accurate, and over time the differences can diverge in some cases.
Is it true to say that:
I can take a TLE for my spacecraft, generate a single state vector for a particular time (perhaps the TLE epoch), and hand it to my friend. My friend can take that state vector and generate a TLE and have it be just as "good" as my TLE.
This particular point is wrong because one single state vector contains no information on drag. You could have a 1000 kg ball of gold and a giant balloon with the same state vectors, but the balloon would de-orbit quickly while the solid gold ball would loose altitude extremely slowly due to its high mass/area ratio.
The way TLEs are made in the real world is a bit complicated. Roughly speaking, predictions of orbits from "fancy" models are first fit to all the observational data, probably meany radar and optical measurements over time. Then TLEs are generated by varying their input parameters in order to best match the fancy predictions.
You can just keep varying each parameter in your home-made TLE, then propagating each time with SGP4, in order to converge on a good fit to the prediction of your fancy model.
But they will never match completely because of all of the "pretty good" approximations in SGP4 that are not in your fancy model, and if you don't use a sufficiently long timespan in the fitting procedures for one model to see how the other model is loosing altitude due to drag, then your fitted TLE will be wrong.