I actually read all the responses and found the discussion really interesting.
I think the bottom line here, as I see it, is that the OP has specific circumstances that are making her particularly uncomfortable with the situation. If it were me, I would address these concerns with my dh. I would explain exactly why I'm uncomfortable with being separated from my child at such a distance, and suggest a compromise - as the pps suggested, everyone going into town together and then splitting up, with the ability to reunite quickly if the baby becomes unhappy.
I'm a big believer in the mama bear instinct. Lots of people would tell me I'm crazy for not leaving dd with a babysitter yet - not even her grandma (MIL) and aunt (SIL, who just turned 14). But I am not comfortable leaving her with them yet and I trust that instinct. There are good reasons for it, but it would take pages and pages to explain them all.
I do think it's odd that the OP's dh seemingly hesitates to go for a walk with the baby or spend time in the backyard sans mama, but wants to take the baby away from the house for an extended period of time. And if the town is 30 minutes away, we're really talking about at least a 1.5 to 2-hour absence.
My dd hates car rides, for the most part, and it would be very traumatic to me to imagine her crying hysterically in the car, needing me, for 30 min while my dh drove home (frustrated with/upset by the crying but unable or unwilling to stop to do anything about it). If I was genuinely worried about that kind of scenario, I would have to go with them. I'm NOT, because I know dh would pull over and comfort dd, give her a bottle, whatever.
We have a very different setup in our home because I work and dh stays home. I cherish the time that I have with dd, but it can be hard to do things in front of dh or spend time with dd in front of dh because I'm already insecure about not being around enough, and if he criticizes what I do (which he almost never does, thank goodness), then I really lose confidence, feel incompetent, and feel like I should just hand dd back to dh.
The best way to feel more comfortable with the other parent caring for the child is to let them be on their own, and find their own way. However, if you have strong instincts and/or good reasons to not be separated at a distance from your 1 year old, I think that's okay and it should be respected and worked with. It's not a matter of saying to your dh, "You can't take ds into town without me," but instead, "I don't feel comfortable being so far away from ds right now. Why don't we go together and shop in different stores so you can find me if ds needs me?" Then, when you do this, do everything in your power to NOT hover or check in with them every few minutes. That would be totally counterproductive and just upset your ds.
Hopefully, your dh will understand and respect your feelings, and you'll do the same for him by providing opportunities (real opportunities) to spend time alone with his son. I think it's great your dh wants to spend more time with his son - you just have to work together to find a way to make that happen so that you're both comfortable.
Have you asked your dh why he wants to take your son on errands? Maybe if you discuss it, he'll tell you things that will give you a better understanding of his needs and perhaps they can be met in a more mutually agreeable way. For example, he might say, "I just want to spend an hour with ds without you running out into the yard every time he makes a noise." That would be totally reasonable and something you might not realize you're doing, that really inhibits his ability to forge a caring, close, mutually beneficial relationship with his son, where his son learns that he can count on Daddy to care for him and comfort him.
Hope you're still lurking, OP!