Let him throw a fit and clean it up himself. No matter how tempting it is, DON'T DO IT YOURSELF!

Shoot... let him huff and puff...as long as HE cleans it up, since 1. they are his kids and 2. he doesn't think they are capable of picking up.

For the record my less than 2 year old picks up daily before nap and before bed. They are old enough. 

Instead of Daddy cleaning up after his kids, why doesn't he just teach them to put one game away before they can get out another?

I understand kids need something to do. No one particularly wants little couch potatoes , whether they be watching mindless tv or playing on their electronics. So I would be encouraged hearing there are still some kids left who enjoy playing real games.... but with that said, parents are not maids. And for those who are good acting the maid role have failed at teaching normal skills of being a participant of the household. How well does Dad think it will fly when Jr tells his one day wife or roommate to 'hey, you clean up after me around here'. Or Sally grows up and loses job after job because she had no organizational skills and was the company slob. 

I mean, the man is correct in that if the kids don't clean it up, it will be him cleaning it up. It certainly won't be you. At least he understands that part. But no one wants to try and get through and around their home al day jumping board games while the kids continue to drag even more out. Hey, it's your home too maybe you'd like to sit down a few minutes and not be surrounded by a chaotic mess... nothing relaxing in that. So if Daddy is cleaning up, could he please stay on top of it inbetween messes instead of waiting until the day is all said and done? 

If you don't teach kids how to be productive participants in a family you will be stuck with them forever or saddle some man/woman with them.

My son is 11, he does what he is asked ( sometimes with a lot of lip) vaccuumes,  does laundry, helps care for pets and makes his own lunch. 

I will be damned if he is not able to function as a grown ass man some day. Your husband is not doing anyone any favors. 

I have told my ex and his wife to have bs help with things, be responsible for picking up after himself. Life is not all about video games and getting what you want.

I had it out with SO about the kids leaving messes.  He gave me some song and dance about how he didn't want to make them clean because their mom made them clean and keep her place immaculate and he wanted them to be able to be kids at our house.  Seriously, he said that.  I went down his throat, and told him he was doing me, his kids and BM a disservice with that stupid BS.  I told him that I deserve a reasonably clean house, his kids deserve to be parented so they'll be productive members of society, not coddled babies, and BM deserves to not be the only "mean" parent who is forcing the kids to clean.  And it's not even cleaning, it's picking up after themselves.  I'm not asking his preschoolers to scrub toilets; I'm asking them to pick up their toys, keep their craft supplies in one general area and not throw their clothes on the floor.  Surprisingly, he got it.  I still do most of the "reminding" about cleaning up, but he is on board and actually OK with them having little chores like putting their dishes in the dishwasher and helping with laundry.  (At the moment their laundry help isn't all that helpful, but it will be in time and they're getting used to helping).

Don’t say anything and let it pile up.

Your H not teaching them to clean up after themselves especially at a young age is a great disservice to them. They need to learn the basics NOW. He should be tired of picking up after able bodied children and its only showing them that they don't have to do it because dad will.

With my biological kids I would teach them to clean up after themselves and when they didn't and left things lying around I would throw it in the garbage. Time would pass by and they would ask about that toy or item and I would tell them its in the garbage since they didn't pick up after themselves and I assumed it was trash. Needless to say my home was always spotless after that. Some may think it was a bit over the top but I got the results I wanted and they are clean and organized. There is never any lip on cleaning up its already ingrained in their brains and its become 2nd nature to them.

While you can't care more than the parent, you can insist on certain standards... especially in common shared spaces in the home. 

So, the rule for the living room might be "one toy/game" at a time per child.  They are welcome to go and switch out what they are playing with.. but the first is put away before the 2nd is brought out.  In their room, they might get a little more lattitude but everything is put away before they go home from visitation.

When he huff's and puff's and says "I will put it away".. I would point blank be telling him that his kids are never going to learn how to be organized and how to clean up after themselves if they are never asked to do it.  That he is crippling them by swooping in and fixing their messes for them...you don't hate them.. and it won't kill them to put a few things away.

My DH is a neat freak but suddenly didn't notice anything when it came to his babies. At first. He eventually got on their cases more than I did. 

First 2-3 years though...I was like. Do you not see the piles and piles of sopping wet towels all over our hardwood floors? The spills in the kitchen and globs of grape jelly all over the counter. The little bits and pieces and skid droppings everywhere? The dirty socks that were taken off while watching TV and left under the couch? 

I swear, I'd watch my skids pour themselves cereal in the morning and they'd hold the box about a foot above the bowl and shake, getting more cereal on the counter and floor than in their bowl. They had horrible table manners and would absent-mindedly play with their food at dinner, crumbling it into bits and making it look like there had been an explosion by their plates (This was mostly SS, though OSD was a culprit too, sometimes. Ironically, YSD was neater than both of the older two). 

And these were not super-young kids. They were 10, 12, and 14 when DH and I got married. 

One time, I swept right after skids left on Monday morning and waited for two whole weeks to sweep the floors again. I swept right before skids showed up and took a picture of the pile. Then, I swept again as soon as skids left and showed DH the difference between 2 weeks with us vs 3 days with his kids. The pile from the skids was at least 4X bigger than the dirt DH and I had created in 2 whole weeks. I told him...this is NOT NORMAL. And, if they're spilling and dragging in this much debris...they need to learn to sweep it up...