ZAU Center Controller reported an aircraft vectored for a Visual Approach descended below their assigned altitude and below the Minimum IFR Altitude.

Narrative

Was working Aircraft X and told him to get the weather and NOTAMs for ZZZ airport. He says he has them and wants to do a visual approach. He eventually asks to go to 3;000 so he can 'overfly' the airport and get a look at it; and eventually shoot a visual approach to Runway XX. Since I had no traffic at 3;000 around the area; I let him do this assuming he would enter a holding pattern over the airport at 3;000. Eventually; he starts descending and I reach out and tell him to maintain 3;000. I issue two altitude alerts because his altitude dips below the MIA to 2;300 I believe. At this point he is NORDO and I try reaching out on guard frequency and also using other pilots to relay and get a hold of him. None are successful. At this point the sector starts to get really busy and I ask for a d-side. Eventually; Aircraft X checks back on; but this time he is climbing to 3;700 even though he was last assigned 3;000. I tell him to maintain 4;000 and to state his intentions; I believe he requests to go to the ZZZ1 airport. He asks for the weather and I tell him to get the current ATIS which is F. At this point his navigation is wildly swinging from different headings; literally 70 degree turns and I ask if it's an emergency; he says negative. At this point people in the area are noticing how busy I am and start mentioning the haze outside. The visibility was pretty poor at most airports in the area but we notice that ZZZ2 airport has 4SM visibility. I suggest this airport to the pilot and he concurs that it would be a good choice to go to. I then clear the aircraft to the ZZZ2 airport via direct. At this time we split off the sector which was incredibly difficult due to the amount of energy and time this particular pilot was demanding. I barely made it through the briefing due to the sheer amount of traffic and complexity at the current time. I wasn't thinking too much about the weather because the pilot wanted a visual approach and he said he had the weather at ZZZ; but it was 100% a factor in this. I think pilot inexperience and ineptitude and the resulting confusion played a huge part in the workload and complexity of this situation. My phraseology probably wasn't perfect regarding the safety alerts but I remembered to spit it out as soon as I noticed he was descending without a clearance. The unexpected actions of the pilot definitely had me a bit rattled and I wasn't sure how to deal with this type of situation as I'd never experienced something like this before; where the pilot goes NORDO and starts doing whatever he wants on an IFR clearance. I think I put too much trust in the pilot and I should have considered the poor visibility outside due to the wildfires. Since this is such a rare event; I've literally never seen something like this in my entire career...In retrospect I should have asked if he wanted an IFR approach considering the visibility and weather conditions at the neighboring airports. I put a lot of trust in the pilot to get his weather and knowing what would be appropriate for himself instead of asserting that he should fly an instrument approach from the very beginning.