Why aren't secondary mirrors offset to get rid of diffraction spikes due to the support vanes? Some kind of ellipsoidal shape mirror could reflect to a secondary mirror which is not in the way of the infalling light. Two advantages are immediately obvious. The primary mirror would not be partly shadowed by the secondary. And the support structure of the secondary would not cause diffraction spikes in point sources.

What are the reasons for why this is not only not the standard, but quite uncommon? Especially on the greatest telescopes, one would think that the additional production difficulties, if any, are not a relevant consideration given the budget and clever engineers behind them.
There is a 1.6 meter off-axis telescope for solar observation at Big Bear Observatory called the New Solar Telescope (NST).  Off-axis is particularly helpful in solar observations because it reduces stray light and eliminates a rotating diffraction pattern which can be severe when the sun is the target.  However, for most astronomical work the additional costs are just not worth it, yet. The extra mechanical costs may be greater than the extra costs for the optics.  The total weight of optics and supporting structures of professional telescopes can be many tons.  There is additional strain required to rotate and support an asymmetric structure like this, i.e. bigger motors are needed.  Galaxies are not bright enough to produce diffraction patterns and the issue of stray light from nearby bright objects is far less of a problem than the foreground stars, zodiacal light, and night sky light actually in the line of sight.  If a nearby star is producing a bothersome diffraction spike, one can usually rotate the field to avoid this. I am not saying that stray light or diffraction effects are never a problem, just that this is not a top problem and, if we need to trade down in size at all for an off-axis design, most astronomers would not accept it. Perhaps if relative costs for these telescopes come down in the future, they will be more common.