Yes, the obvious question is: are you riding hard enough, even for short bursts? However.....

Unless the recent research has changed, cycling is slightly less aerobic than running, simply because the upper body isn't really used.

extra 'however'....
I've never, EVER found cycling has hard (painful) or breathtaking as running. It's always been more about the legs than the lungs for me; like my lungs always have some ceiling room. The only time I get breathless is doing full-on sprints and some short intervals, and even then, I get my breath back pretty quickly.

When I compare this to the 'old days' when I used to run, then played Aussie Rules football before that: ugh! Pain! For memory, 400s and 800s were the worst.