There is no such thing as "over-kill" when the intent is varmint eradication. Only if you plan to keep the pelts, does "over-kill" belong in a sentence. Regarding the three you listed, .243 Win, .270 Win and 7mm Rem Mag, all three are effective and efficient.
The .243 Win is a perfectly suited round, loaded with 75, 87, 90, 95 or even 100 grain loads. However, I assure you, that if you load a .270 Win with either Speer 90 grain HPs, or Speer 100 grain Hot Core, there won't be much coyote left for postmortem examination. Load them to 3,800 FPS and it's like a nuke hitting the poor critter. If I were deploying my 7mm Rem Mag, I would handload with 120 grain Nosler Ballistic Tips. Take your last look at the sorry critter before you pull the trigger, 'cuz they ain't a gonna be time to visual, nor much substance remaining for evaluation after the shot! Either the .270 Win or the 7mm Rem Mag, when loaded as described, will literally create "red-mist" upon impact. There will be left behind the coyote, about an 8' to 10' radius of former coyote.
For me, if I were going tomorrow, I'd gin-up some 100 grains for my .270 Win and never look back.
Geno