We have the Cruz and have checked it a lot without problems. I put a department store suit bag over the seat part - it fits well as long as you fold the footrest.
We also have a Chicco umbrella stroller for shorter trips or all trip now that our son is bigger. It was $100 (open box on Amazon) and works great.
Recently though I've seen a lot of these micro strollers for travel that I wish I knew about before... Here's one :
https://www.worldshop.eu/en/product/Cybex-GB-POCKIT-buggy-sea-port-blue/1751541?p=r9eNYY_gS2g