Good luck with setting large hooks with a normal Avon rod.
Carp rods with a good through action will be fine, tippy SU rods may struggle. And, if you are fishing with 8" + deadbaits (plus the rig) Avons or Barbel rods will be challenged.
Pike fishing is no different to any other form of fishing. For example if you want to go trotting, people will swear by this rod or that rod, but actually any old float rod will do a job. If you want to get the most enjoyment out of fishing a particular technique, or for a particular species then using a purpose built tool will of course enhance that enjoyment, even if only marginally.
I fish for pike a lot. I get limited enjoyment heaving out a herring 70 yards, or sat behind a battery of rods for hours. What is really enjoyable is fighting a good sized fish on capable, but balanced tackle that gives the angler the best enjoyment, the best chance of landing his quarry, and the least chance of damaging the fish, directly of indirectly.
Hook up to a decent pike in a fast running river on undergunned Avon rods and you will very likely lose the fish in snags where it may perish.
Just my view.