re: The planet's orbit might be highly elliptical, too. - also a possibility.
Highly elliptical planetary orbits tend to induce shorter-lived, but higher radial velocity pulls on their primary stars, so these are the ones (all else being equal) most likely to be discovered first using spectral Doppler techniques. So even if these highly elliptical orbits are rarer, most of the discoveries will feature this type. This is what astronomers call a selection effect: the phenomenon that seems most common is not, it just seems that way because it is easier to find.