Reproduction does not lead to cancer, there's no connection. My oldest breeding doe is 8years, smaller litters, but no birth defects or an unusual amount of stillborns. It can be more difficult to get a doe bred the first time when she's older than about 1 1/2-2 years, depends on the individual rabbit, even 5 yo get happily bred if you absolutly do not want them to.
What's right is that false pregnancies can occur, that is something that is hard or disturbing to watch, but usually doesn't hurt the rabbit (some get it real bad so it can become kind of stressful for all involved). Some good reasons to spay pet rabbits are connected to this topics, but I guess that's not the question here, doesn't sound like anyone would fork up the money for that.
Starting with the wrong arguments isn't the best way to convince someone.
So, what would I consider valid arguments? You mentioned an abundance of rabbits, bring free market into play and you get low prices. Unless those rabbits of yours are pedigreed, show quality rabbits there simply is no money to make. And even if your rabbit was expensive (that only says that you paid a lot for it, nothing else), you do not run a rabbitry with good reputation, so nobody would pay much no matter what.
As you said yourself, too many rabbits out there, all those people who thought like your mom, thought it is just cute to breed their bunnies, or simply had one of those very common "accidents" ruined the market.
It's actually pretty hard to make a rabbitry pay for itself. I charge 12-15€ for my mutt rabbits, can sell max 20% of the offspring in 5-6 months.
Not sure about the zoning laws in the US, rules of city or neighbourhood, your country is chocked up with so many rules and laws on so many levels that sometimes it's hard to tell how many animals are allowed - one big litter could be too much in some places.
Also, raising rabbits isn't for free. Food isn't for free. Vet isn't for free ( Enough money should be set aside for any kind of emergency, and what I heard vets aren't cheap over there). Cages and space aren't for free. Young rabbits are not litter trained, so there's quite some cleaning work involved. Has your mother bred rabbits before?
There's always the possibility that you can't sell all rabbits in time, what to do with those? More cages and supplies needed.
Also, breeding sometimes sometimes leads to difficult situations and decisions one should be prepared to deal with.
I make a clear distinction between lifestock and pets. My breeding does live outside in hutches and dig up the backyard, my two pet rabbits share my house and garden with me, no cage or any such thing.
If this is your pet rabbit, that you care for, your responsibility, without question, in my opinion it would be a violation of your rights and you can say No. If your mom does most of the work, or this is the common family pet, well, than it would be more up to discussion. I would try to convince or involve some allies too, but I wouldn't spread the argument to everyone.
