You need to nip this in the bud now, because otherwise you're going to have it hanging over you, and despite your DH's waffly "oh i'm sure it'll be fine/not happen", that doesn't really help with the ambiguity going on.
For them, and you, you need to make it 100% clear, politely, that this won't be happening.
"Hi, just to confirm after we discussed school applications this week/last week/whenever - to be clear: we won't be sharing school runs with anyone and we're making our own arrangements xx"
Just write it clearly, without any room for "but what if"... nonsense.
OP, stop dithering here and just cut it off at the root.
If you and DH need to work on boundaries, this is a great time to start.
And TBH the relationship doesn't sound healthy or one i'd want to encourage for my DD so i'm wondering why you're even keeping it going - it's sometimes not enough that your DD might want to keep a friend that she gets on with. Your job as a parent is to manuver her through challenges and develop her resilience, but at such a young age, reducing contact with children who aren't a good role model or influence is a good move - i'm wondering if you're such a people pleaser you can't think clearly about WTF you're doing here. (I mean that nicely, i used to be a people pleaser until it absolutely broke me, now i realise that being seen as "rude" is mostly just "other people walking over you".)