@VesperLynne
To be honest, yes it can. I was 8 when my parents split up. We had to move from where we lived to a new school and away from my friends. I was bullied mercilessly at my new school and nobody seemed to be able to stop it so I coped as best I could. Eventually , after I was kicked in the face and lost my front teeth, my father arranged for me to go back to my old school. I left home at 18 to start my medical training and I never went back. I've been very happily married for 23 years so I guess it all came together for me in the end.
This is awful, but not an inevitable part of parents divorcing. Bullying can happen at any school for any reason, not just to the kids of divorced parents or those who have to move schools.
OP I divorced when my DCs were between 6 and 12. It was obviously hard for them to adjust to the change of their dad not being here all the time, but one of the main reasons we divorced was that he was never here anyway, worked weekends, went out on his own on days off etc so having to actually spend a bit of time with them once a week was more than he’d ever done before we split!
We stayed in the family home and he got a place not far away at the start. The upheaval of remembering to take everything they needed on a school night was probably the worst part for them - they had to take something to do, home clothes for the evening, new uniform for the morning, the books they’d need for the next day plus PE kits etc so it was a bit of pain for them to have to think ahead, but taught them to be organised when they missed a couple of things sometimes and had to pop back in the morning to get them!
These days they usually just stay with him if it’s a weekend night, or he takes them out for dinner and drops them home afterwards, so it means I don’t really get much time to myself, but they’ve grown into very helpful and self sufficient people so I’m not doing it all alone anyway. If I go away for a few days with my DP, the ex will come and stay at my house with the DCs to make it easier for them. He also fixes things while he’s here, so I often come home to find lightbulbs replaced, doors fixed, the oven cleaned etc
and I let him store stuff in my garage and offer for him to stay for dinner sometimes, so we’re both very amicable. I even bought wrapping paper and gift wrapped a load of things he’d bought for his family recently as he knew I’d make a better job of it 
Being on good terms helps hugely - problems start when one partner is abusive or unreasonable and then you’re at war over everything. Boundaries are very much in place, but there’s nothing awkward or uncomfortable here.
Their dad had a GF at one point and my DCs adored her, she loved Harry Potter so it was nice for someone to share that with!! But he’s been single for a few years now.
I have a DP, who’s fun and kind, he’s not perfect and there are ups and downs as in any relationship, so we don’t live together full time, but he brings so much good stuff to the table that the DCs love having him around too. I wouldn’t put them in a position where they had upheaval over my relationship again, so no plans to move in together until they’re all grown up.
If you and your H can both put the DCs first in whatever decisions you make re access, housing, partners etc there’s no reason they will be “messed up” by divorce - that’s a really nasty view on separated families.