You know him and we don't, so only you know if this might work, but...What f you tell him most of what you posted here (you put it pretty darn well when you talk about how the service makes big differences that he does not actually see) and then ask him to give you this service as your birthday and Christmas present for the year?
Seriously, I would say, "OK, we need to agree to disagree, but I am asking that you take me and my concern here seriously, and let me have this one: Consider it my present. You will not hear one peep from me about wanting 'something to open' and you will not see so much as one raised eyebrow when everyone else is opening stuff at the holidays. Instead you will see a more relaxed wife who has not spend the preceding year herding kids and deep cleaning too."
Should you need to do it that way? Well, no. He should be more open to respecting your opinion on this since you, not he, spends the most time in the home. And this really does come down to respect. But if he can't manage that, then see if he'll go for the gift idea. I don't love it, but it might at least get you the service back.
Alternatively, have you ever written down for him EVERY single thing the cleaner does at each visit? In excruciating detail? With the time it would take for you to do each of those things? Put it in terms of how many minutes each week/day you would spend on the same things. The cleaner does it faster, that's for sure. He may need it laid out for him as a business proposition: If I spend X minutes a day, Y hours a week, on tasks A, B and C, that leaves this many fewer minutes I can drive the kids to their dance class/help at a volunteer thing/participate in school activity/etc.
Of course, if he says that school participation by parents, or volunteering at a kid's preschool, or getting kids to activities doesn't matter -- then you have a big red flag that really nothing you're doing at home and with the kids is as iimportant as money or his own job, and that's a very serious discussion that goes far beyond the piddling $80 cleaning service. I hope that's not the case and he's open to reason and to showing you some respect.
By the way, like Ziggy below, I am sorry to see some posts judging you and implying that you shouldn't need a cleaning service if you are at home with your kids and your kids have any kind of care or school. We don't know your situation and it's rude of others to judge you. I can barely keep up with ONE teen child who is old enough to help out - my hat is off to you for keeping up with five, and if a service really makes a difference in your quality of life (and your husband's though he does not see it), then push to keep that service.
I also think the idea of leaving stuff to get gross so he learns to appreciate the service is, well, passive-aggressive. I would first much rather talk to him adult to adult, when the kids are not going to interrupt, and present it as a business proposition of stressed wife versus what is actually a pretty cheap cleaning fee. And don't forget -- it's about his respecting your opinion of what's necessary in your domain. Passive-aggressive tactics don't get respect but only make people angry.