You can dead-head if you want but try to leave as much foliage as you can as that helps to increase the rhizomes (ie., treat them like you would do bulbs).

There are some who have reported a tradition of mowing them to the ground every year after bloom and they bloom fine the next year but I expect that's done with old established clumps.

I am surprised by your description of your peonies as "making such a mess" after the spent blooms have been removed. Mine are quite attractive green shrubs until they die back in the fall. Al

I have some that become a mess after blooming and some that remain beautiful shrubs after blooming. I tend to place the ones that become a mess, behind the ones that remain as beautiful shrubs. I try to wait as long as possible before cutting them down, but I have one that is just an eyesore right now. I'm resisting because the seeds aren't ripe yet and I'd like to collect them this year.

I dead-head asap when done blooming but leave the foliage until fall. The plant needs the foliage to 'feed' the roots. DO NOT leave the foliage on your peonies over winter.