These sorts of things might be a case of trial and error, and it's important to consider that what works for a while might need changing later. However, we used a stroller for our daughter for just over ten years and always found it invaluable when getting through airports.
It took a few trips to figure it out, but provided we stuck to certain rules we actually found the stroller helped with everything else. First, we always made sure she was in it (the only times she wasn’t was going through security and during boarding) and one of us would push her while the other carried her car seat (we’d swap over if we needed to). As long as we kept to that routine, we had the peace of mind of knowing that if we had to talk to airline/airport staff, figure out what direction to go, get stuff from our bags, etc., we were able to focus on that knowing she was comfortable and safe.
Second, because it was not always a popular decision, we were very serious about explaining that it was best, and always made sure she had a toy and when she was older a book (either of which could quickly go into a carry-on bag). This was also important when we were trying to find somewhere to sit, as stroller acted as an extra seat so we only needed to find space for ourselves if the gate or seating area was really busy (though arguably this is also true for a car seat on a cart).
Moreover, having the stroller with us for as long as possible reduced the chance it might get damaged or lost. By gate checking the stroller, we could pick it up immediately after leaving the plane, and then we'd simply repeat the process and she would stay in it while we went through the airport which made things much easier, especially if the airport was new to us.
As GUWonder has pointed, it's worth checking with airlines what their stroller policy says, but if our experience is anything to go by, we’d suggest taking the stroller and gate-checking it because for us it made things easier, quicker, and safer. It’s different for every family and child of course, so be prepared for it taking a few trips before you know what works best for you. We were lucky that we learned fairly quickly what worked for us and that our daughter naturally travels fairly well, but setting up a routine was the key and the stroller was a massive help in doing so. Once we'd figured it out, it became clear how much it helped us and our daughter was more than happy for it to be a consistent place for herself to be in her own world; by about 8 or 9 years old she would buckle herself in the stroller and just carry on reading as if the whole flight hadn't even happened!