Exactly. If one is playing multiple genre's and styles, then one should consider that a "workhorse" may not serve all those applications with any degree of authenticity.
Having several "one trick ponies" on the other hand, will serve them wonderfully, and playing on multiple instruments with different scale lengths, fingerboard radiuses, pickup voicings, timbre's, can only improve ones tonal and playing vocabulary. Gosh, look at all of the gear Carl Verheyen, and the multiple pedal board/amp configs he has for studio work. Case in point.
While a cover band may not be able to practically lug around multiple amps, perhaps the amp modelling route may work for ya, or even the Variax + amp modeller setup. I know about all of the cork snifferey around it, but most cover bands that are raking in any amount of cash have a fan base that isn't consumed with wether or not you're using modelling technology or old school gear anyway. The investment may help you achieve the sounds you're looking for, while minimizing your stage footprint, and expedite your setup to fit more songs in the set.