Choice of launcher seems to be an early decision in the design of spacecrafts. There would be an advantage to having the option to easily change launcher, especially for an interplanetary mission with a rarely recurring launch window, in case the intended launcher is grounded for a long time. A mission prepared to deal with an emergency, such as a space tug to save a failed orbital insertion of a crewed or otherwise valuable spacecraft, or for last-minute deflection of an asteroid, would also benefit from being able to launch from any out of several frequently launched rocket types (replacing the payload of the next one prepared to go).
The most frequently launched rockets nowadays are Atlas V, Falcon 9, Ariane 5 and Proton (assuming that Soyuz is too feeble). Incidentally, two of them have been grounded for 4 and 7 months now.
Could one design a spacecraft to go with at least three out of these four?
Would it need a bulky adapter stage?
Could one make adapters that could take almost any satellite or spacecraft on any of the launchers? (To steal the competitors' customers)
Proton is horizontally integrated, the others vertically. Is that outright disqualifying for inter-launcher payload compatibility?
I suppose one "could", and that it comes down to how much would it would cost in mass and extra engineering. If it would be cheaper to have a dedicated launcher on some level of standby instead.
Delta IV and the Japanese H-IIB are AFAIK both based on Delta, so they might be quite compatible. However, they are the most expensive launchers and do not launch very often.