Depends.
If a child’s left at home with one parent, it’s not the same as them being left at home alone. Provided they take full responsibility for supervising and safeguarding them, i.e. could live with themselves if the child came to harm. Lone parents have to do this all the time, even if they’ve been kept awake most of the night!
Obvs parents can’t be expected to work while supervising young children, other than for a short while now and again. In that case, it’s best to have the child in the room with them, so they know where they are and what they’re up to. Young children can be unaware of dangers (even ones they’ve been warned about) or do silly things when playing, because they’re absorbed in their play, e.g. putting things around their neck or in their mouth.
I’d wouldn’t have left mine completely unsupervised (i.e. someone fast asleep) at home, even for a few minutes, until late primary age, e.g. 8 or 9. However, I left them with their dad at every age, even if he was in bed, as he’d take responsibility for being aware of what they were up to.
It seems a little unreasonable for your DH not to offer to supervise for 5-10 mins, to save having to take your DC with you for every drop-off or run to the corner shop. When you have young kids, sleeping in is a luxury that’s usually only possible at weekends (if you take turns) or if kids are elsewhere. Again, lone parents have to supervise alone at home all the time, so 5-10 mins should be manageable.