The honeycrisp I planted two years ago was a janky cheap sitting in front of the grocery store in a pot seedling, so who knows what rootstock it’s on or if it’s truly honeycrisp, but I’ll probably find out this year. Not that I can taste it, honeycrisp make my throat swell up. So I’ll have my kids taste it and I’ll graft a bunch of other stuff on it, frankentree style. I just got some multiple scions each of Calville Blanc D’Hiver (1600s France), Niedwetzkyana (red flesh, 1800s), Rambour Franc / Summer Rambo (1536 France), and Lady / Api (1600s France). So I will graft those on my neighbors’ Gala I planted maybe (it’s leafing out) and on my honeycrisp if it seems more bud swell-ish. And maybe graft some on my friend’s giant never pruned definitely not dwarfing granny smith. Or maybe I’ll wait for some until my rootstock order from Cummins arrives. I’m worried my honeycrisp is on a full dwarfing rootstock and may not live very long. If I make some frankentrees I’d like them to have some longevity.
This is a classic Bradford pear experience. LOL. Sorry to laugh but I am laughing with you. They are a controversial pear specimen through and through. I agree with @MaartenFoubert, to a degree, and I agree with you and your experience in this case. In theory, they are a wonderful and prolific nearly wild growing (once established) theoretical root stock specimen. BUT, on the other hand they are just not very durable and they can root shoot stoloniferous suckers. They are fairly weak trees in any adverse weather events - splitting and falling over is entirely normal. This just doesn’t bode well for specific intention rootstock use. However, in a large acreage as a novel project I continue to think they would be a worthy candidate for a massive pear Frankenstein population. IF I had the time and the hands and the acreage I would consider it because I am a glutton for absurdly low-probability fruiting projects - ergo I continue to work with a 48 N latitude Asimina triloba project (I am even further encouraged after this past winter! Gods be good!). Back to the ‘Bradford’ pear…for me, my main issue (a rather humorous one) was also the odor of wafting semen (!! I’m serious) when they established a larger population. I had a hard time with this in Missouri. I laughed at it, initially, but then I couldn’t get this wild smell of sperm out of my nostrils. 
Maybe I’m crazy. I don’t know. Anywho, where I tend to direct my thoughts in the end is the nursery trade uses specific rootstocks for many reasons and they have entire branches of their field dedicated to breeding and trialing specific rootstocks. I learned my lessons the hard way when I dedicated 7-8 years of my life working with a fairly significant population of apples I had gathered in and around the Willamette Valley of Oregon for two winters. I had a novel idea of grafting them to the native Pacific Crab Malus fusca. A significant number of the grafts took - I was overly thrilled. The thinking was this specimen of wild crab is extremely hardy and can withstand flooding and drought. BUT, over the years many a problem arose within the population. Grafts failed over time. Bark split over time. Then I had a filed graft union and a suckering wild crabapple population. In the end, I just planted the remaining trees in a large block and essentially told them “If any of you want to live and have what it takes, I give you that opportunity.” From that point onward, when I wanted to work with grafting any materials I simply reached out to nurseries I trusted to get their most recent thinking. With apples, in particular, almost all the heritage varieties I know of love an EMLA-111 semi-dwarfing rootstock. It’s a great marriage. Anywho, happy fruiting everyone.
“Yeah the issue for me is when the bradford crosses with a different pear and spreads invasively choking out native plants”
If a Bradford pear crosses with my awesome pears, the subsequent pears may have bigger berries, too big for birds to eat, so they stop eating the berries and the spread stops, so my pears are actually stopping the spread.
"they KEEP sending up shoots two years later, probably from tiny bits of root. "
Nature is telling you to stop growing apples and graft on the pears 
Join the dark side, we have more fun
Maarten