Can we more accurately predict orbits or measure position?
As a general rule, orbits, by far.
The state of a planet, asteroid, comet, natural satellite, or human-made spacecraft has six degrees of freedom: Three for position, and three more for velocity. Orbital elements similarly have six degrees of freedom. Orbit determination mandates having a good fix on all six degrees of freedom. As determining position on the other hand requires only having a good fix on the three positional degrees of freedom, it seems to be contrary to say that orbits are more precisely measured than position.
Determining position requires accurate measures of distance (range) and bearing (azimuth and elevation). This typically is not possible as a one time measurement. Optical telescopes only determine bearing; range is difficult if not impossible with an optical telescope. The Keck Observatory claims an accuracy of 1 arcsecond, which at 1 astronomical unit (AU) distance corresponds to an uncertainty of 725 kilometers. The transmitter / receivers used by NASA's Deep Space Network have a half-beamwidth of about 0.017 degrees, which at 1 AU corresponds to an uncertainty of 44000 kilometers.
On one hand you have the hard problem of measuring exact position and velocity at light minute distances.
That it takes time for a signal to be transmitted from an antenna to an object in space and then be reflected back to Earth is a feature rather than a flaw. Time and frequency are the two things that humans measure most accurately. Modern measures used for orbit determination, particularly for human-made satellites cruising between planets, forego using bearing (azimuth and elevation) because those measures are lousy compared to the range and range rate measurements. Range (radial distance) measurements can be accurate to the sub-meter level by measuring the time delay. Range rate can similarly be measured to high precision via at the doppler shift in the transmitted versus received signals. Human-made spacecraft that go beyond the Earth-Moon system are outfitted with equipment that augment range and range rate measurements. However, even natural objects benefit greatly from range and range rate measurements made via radar astronomy.
Precise orbit determination relies on having captured dozens, if not hundreds or even several thousands of these imprecise and incomplete measures. Statistical techniques coupled with physical / mathematical models of orbits reduce the uncertainties and fill in the incompleteness. Orbit determination is much more precise than any one measurement, especially when used for interpolation.
Extrapolation (e.g., future predictions) remains a challenge. One of the biggest challenges for Near Earth Objects is the non-gravitational forces such as solar radiation pressure and the Yarkovsky effect that act on such objects. Is a NEO a potential hazard or not, and if a NEO is a potential hazard, when might it hit the Earth? Those non-gravitational forces make predictions a bit murky. Assessing these non-gravitational forces is one of the key science objectives of the OSIRIS-REx mission, which has already collect a sample from Bennu and is intended to rendezvous with Apophis in 2029.