Take half the toys & games and put them in the attic or basement. Bring them out in 6 months and they will feel like new. You can also have a "rainy day" or "blizzard" section - you don't want to be in the attic all the time, but on special occasions, you want the option to haul something out. Otherwise mark your calendar for 4 months or 6 months - you can switch things twice a year or 3 times, no more. Once a month, as you suggest, is going to be a lot of work for you if the toys are somewhere else, but that's the only way to really cut down on the sheer volume of stuff you have.
Your kids are telling you that they have too many choices and can't do the thought processing associated with deciding, so they go for the same standard toys every time. Keep an assortment of different-themed toys and things that use different skills (you are already an expert on things like gross motor, fine motor, deductive, auditory, visual, kinesthetic, etc.). Be sure there are different things for each child because their needs, development and interests are different.
Donate anything you think they are really done with. Have them go with you to a social service agency that accepts used toys and gain the experience of donating.
Make sure boxes are labeled on the side with a photo of what's inside if it's not otherwise obvious - you don't have to take a picture - just print out something from a website or cut out a part of the original box, and paste that pic on the side. You can also attack something to the canvas drawers you already have.
Use storage bins or rolling carts if you find them at yard sales or the dollar store. We just used Rubbermaid-type bins with handles (for dragging out of the closet) and well-fitting lids. We had some milk crates for larger items that didn't fall through the openings, and it was easy to stack the crates and see inside. (Later, he used those when he went to camp for an easy-to-assemble bookcase.) Again, put a photo on the side (not the top - they can't see it when things are stacked!), which lets pre-reading kids figure out what's where, and it also helps the ability to clean up logically. "Clean up your room" is too general for kids, but "put all the cars in this box" is much more manageable. Mix things together in a logical way even if they weren't purchased together - all the Legos go in one bin, all of the train tracks and cars go on another one.
I don't know if this would work for you, but we had a handyman build a large table for us, and it was a place for our son to set up all his trains. We had it in the basement. We started with the Brio and Thomas trains, and he added in his own Lego structures/creations, and some other things he got from an older friend of the family. He mixed in K'nex and Construx (not sure if the latter is still around), and had a huge set-up. As he grew into Hot Wheels, the Brio stuff got sold and he moved into cars and race tracks and so on. We put the Rubbermaid bins underneath the table, but he could slide them out whenever he wanted to. The handyman made 2 sets of legs so we could raise the table up when he got older. I don't know if you have that kind of space, but it's a great way to have a big place where stuff can be spread out and not picked up every day, but still not be in the way. Mixing toys from different sets helped build his creativity and the sophistication of his creations - maybe it's just a coincidence, but he's now a civil engineer employed by a big construction company - and it all started in our basement!
Good luck.