In the Netherands and Belgium Wilson speakers are often demoed using Ayre electronics. To me the Wilson-Ayre combination always sounded good. However, this is certainly not the only good combination. About two years ago I almost bought a pair of Wilson Sophia 2. Back then I have demoed the speakers for a few days and drove them (due to budget limitations) with the Musical Fidelity A5 integrated and cd player, and also with the cheapest integrated and cd player form Ayer (serie 7 I think). Both combination were great. Some may argue that the Sophia deserve better electronics. That is true. However, my point is that you do not need top of the line electronics to get great sound out of these speakers. I have also demoed (about a year ago for an hour or so) the Sophias 2 with Accuphase electronics, i.e. the E-550 and E-450 integrated with the DP-500 cd player. Again with great results. Also, from what my dealer was saying Mcintosh, Pass Labs and Naim electronics also work great with these speakers. Thus, if you already have a decent amp do not rush to replace it as these speakers are not very peaky (like you said). Try as many amps as possible and wait for the best bargain.
Good luck!