Having grown up with MPP's (multi pump pneumatics) like the Crosman 2100, Daisy 880, and Crosman 760, I will say they are plenty accurate and plenty powerful for pests within 45 feet. They also have the advantage of being multi power, down to the level of shooting salt at flying insects, or even using wax bullets to snipe a flying pest that has landed in front of one of your windows. Even more useful at times, is an empty chamber. A puff of air at point blank range will take care of any crawly - e.g. that silverfish in the corner of the ceiling. I even discovered that you can load plastic cap gun caps in a 177 MPP. 4-5 pumps, and the cap will explode on impact with a hard object, out to 15 feet or so. I also discovered you can shoot a strike anywhere match at a brick wall, and it will light.
MPP's are the 12 gauge shotguns of the airgun world - open cylinder and sawed off, anyway. Dirt cheap, reliable, and at short range, you have a wide range of ammo and power options and just enough power for dispatching pests large and small.
Having had a Benji Silver Streak in 5mm, I gotta say they are noticeably more powerful than a Crosman 2100 or a Daisy 880, but man are they a heck of a lot harder to pump and a heck of a lot louder to boot. To me that's the obvious crossover point where it makes more sense to go with a springer/gas-piston for even more power but much less noise and effort.
CO2 is just too inconvenient and expensive for me. In my experience, you can't leave the CO2 in the gun. It's shoot it or lose it. It makes sense to me in a compact pistol format, and I had loads of fun with these kinds of guns as a child, but I don't have any use for one, anymore.
I'm pretty late to the game with springers, buying my first one 4 years ago. But I can immediately see why they're so popular. It's way easier (at least for a normal sized adult) to cock an average springer than it is to pump up a MPP. If you have the arm length and leverage, it is almost as easy as breathing. And you have to load a pellet anyway, so why not cock and load the gun with the same motion? I found out you can burn through a heck of a lot of pellets with a springer once you find a comfortable spot to shoot in. Accuracy is plenty good and not that hard to achieve, IMO. It's so easy to cock my Venom that I think someone (everyone!) should make a repeating breakbarrel or side-lever rifle using a rotating magazine. For me, it's 10 times as much work getting the pellet in the barrel as it is to cock the gun.
The biggest drawback for springers, to me, is that they are full power, only. And there's no fooling around with light-weight creative loadings, because that will damage the spring. So there's always a place for an MPP in my stable.
PCP, I have no experience with. I think they're cool like a silenced 22LR would be cool. I just haven't passed the money and convenience threshold to want to get either, yet. I can pick up a 22lr or a springer and have 100+ shots ready to go, on the rifle, with another couple hundred rounds in my pocket. Spare ammo can be stored in a much smaller space than an air tank and hoses.