You're absolutely not overthinking it; this can and has been an issue, or potential issue, for blended families I know. Success of each approach is largely dependent on the Dad's behaviour and view on it.
If the Dad doesn't make it a big deal, the DSC won't either.
Another issue you specify regarding clubs, will depend on the child's parents attitudes to it.
My EX-DSCs Mum refused to take EX-DSC to swimming lessons which me and my EXDP paid for, during her weekends. This meant that every Saturday morning EXDP was committed up to 12 midday. He'd then say he was too tired to take our DC anywhere following.
I ended up taking our DC to all their extra-curricular activities by myself as my EXDP wouldn't as he said he was tired from ferrying around his DC, even on the weekends they weren't staying with us!
You haven't mentioned what your DH's view is on all this, but he should be the one being proactive about making sure all his DC get equal support from him, and explaining the difference in households. You will be the bad guy if you are having to do all the talking.