You absolutely should not move in my opinion. The biggie is that you don't even want to stay in the area for more than a few years - and for that you want to do the hard work of moving (with two small kids) to a different nearby place? That then needs to be furnished? With stuff that then needs to be moved out of the area? Umm, no.
But there are a bunch of other things as well that jumped out at me:
I agree we might need more space as kids grow but I'm perfectly happy with our current setup since we cosleep and use the other bedroom for play, guest room and other uses. I want to keep a small living space as much as possible so it's easy to clean and maintain. But I also feel kids would love to have a bigger place so they have fond memories like we do about our childhood homes.
Wait, what? You do know that kids' memories depend on whether their childhood was happy, not on the house size, right? You both have happy memories of your childhood, so of course your childhood homes play a prominent part in that - but not just any big house, your EXACT houses, just because they were yours! I promise you that similarly happy memories can be generated by an apartment :)
Even the house we have, we spend most time in the living room/kitchen and barely use the bedrooms except sleep/nap during the weekends. We just love being together than each doing their own thing.
So, why do you need more rooms again? Your kids will want to co-sleep for a few more years (and if not they can/likely want to sleep together in the kids room) and by then you want to move away anyways.
* I do get a bit jealous when I see lovely home pictures in Pinterest or when friends share on Facebook.
Nope, sorry, doesn't count as a reason!
* My husband is handy but hates doing things himself. He'll do chores and help with stuff if he thinks it's needed but his bar for something being needed is very high. I don't want to get into the position of having to nag him to take care of things. That said, another part of me thinks I need to give him that chance - if he doesn't do it after moving to a single home, that's one thing. But he might surprise me and start doing things himself (since he has more ownership there than with an apartment) which would be awesome!!
I don't even understand this. Give him a chance to do what? Do stuff he doesn't enjoy? I get that you want him to be more hands-on but it really doesn't sound like he is at all interested in doing it himself, so you give him a chance to... conform to your wishes?? Plus why would he feel more interested in it when its a rented house as opposed to a rented apartment?
* I sometimes feel we don't entertain as much since we have a smaller space (toddlers love to run around) and that has dampened our social lives as more and more friends start having kids. But I'm not sure if it's because of the kids' ages (easier to stay home with toddlers and babies than visit) and parents just being busy or if it truly is a space issue.
Yeah in my experience it's more about the age/busyness, but YMMV.
* As kids grow more active, even letting them just be in the backyard while we relax is easier than getting everyone ready and out the door to the park a block away. And I can get cooking or chores done while they get some nature time (instead of being cooped in watching TV or wanting to be engaged).
I think it will take 4 years (at the very least) until you will be comfortable having your not-yet-born kid outside without supervision, even with a fenced-in garden (they can and do climb fences from much younger..). At which point you want to move anyways!
Full disclosure: we moved to a larger place earlier than we wanted, when the kids were two and four. The new apartment happened to open up in the best house in our area, and literally checked all the boxes of how I/we wanted to live. So we took the opportunity! But we also plan to stay there until the kids leave.
Before that we were living in a 2-bedroom place and like you we didn't even really need the second bedroom except for storage. It was an apartment complex with an associated playground, but it wasn't fenced it and you had to go around the building to get at it. So I also thought a fenced-in garden would be much more convenient - but now I might wait until the younger one is 5 until they are allowed to be by themselves in the garden (I just don't trust them to not explore the neighboorhood on their own!).
Oh and regarding the kids happy memories: for months the kids always wanted to move back to the old place (to our adult eyes much less desireable), although everybody assured us that kids this young forget about it in a few days... even now they still sigh about the "pretty kitchen" in the old place! And about those more rooms, well, they still refuse to sleep without adults present, and after that phase I expect they will want to room together until 8+10 or so. At which time, again, you guys will have moved to another area anyways!
TL;DR: Don't move!