Actually, it's a car seat that should never be counter-checked. A stroller is okay. A stroller is not a key piece of safety equipment like a car seat is. I would check the stroller as luggage, put the car seat on a luggage cart, pull it behind me (older babies can ride in it) and then use the car seat on the plane. For those of you who fly with lap babies, ask about an open seat and if not, then gate-check the car seat.
I rarely see big strollers here in Europe but the sidewalks here are narrow. Usually when I do, they're tourists! I also had a large double stroller but only used it to go to local parks (the only place I saw big strollers in Europe) but never traveled with it. Most of the double strollers are tandem, like mine was, not side-by-side, which I wouldn't recommend here. I was surprised how many times even a small stroller wasn't allowed places when mine were little. Museums, buses, etc. Why I used the sling so much here!
I'm surprised that you were able to travel with a Bjorn. So uncomfortable! My shoulders would be killing me after 20 minutes (must admit, big babies!) and it was in the closet at 6 months (never used it again!) I found a ring sling or wrap was better and allowed me to discretely breastfeed (I don't recommend any "hooter hiders" in Europe). A good quality baby carrier should go to 2-3 years of age. I usually recommend any baby carrier that hangs the baby by its crotch. A lot of people, especially with older babies, get a front/back carrier. Toddlers love to be carried on their parents' backs!
If there is a very long connection, then it's a good idea to have to stroller to have somewhere to put a sleeping baby down. Otherwise, it's actually easier without gate-checked items, like you said. Of course, you have more time to wait for gate-checked items if you have several hours in the airport. If you have a lap baby and have to wait for a car seat to come up from the gate, you might as well gate-check the stroller too.
Many European airlines let you gate-check the stroller but don't return it to you until your final destination. Why I didn't fly Air France for years! I just couldn't see negotiating Charles de Gaulle alone with three kids and no stroller (I don't live in Paris and have to connect).
I usually didn't have the luxury of taking the Paris metro with my husband and it was totally do-able with one person. I would wear the baby (usually in the sling) and the key was to have a one-hand folding stroller. You have to be able to have it pop down and can be picked up so just wanted to add that tip, for those who can't travel with another adult on European public transport.
But funny how you found the same solution I did, putting the stroller outside the paid area and using the turnstiles normally to exit (or the opposite). In London, they were good about opening the gates. Paris, less so, and that method worked well. I think the risk of theft is minimal. The strollers stolen in France that I've heard about have been left in parks or on the ground floor of apartments (one girlfriend was tipped off that a homeless guy was using her twins' double stroller!) It's unlikely that they'd snag it in front of the parents (and if so, how fast could they go??)
Glad your AirBnb worked out. I don't use them after an owner cancelled on me without notice. Be aware that they can do it. I complained and was ignored. I wasn't the only one he did that to and he's still renting another place in NYC. Just be aware that you can get off the plane somewhere and they can say "sorry!" and you have no recourse whatsoever!