I'm rarely at a computer in the airport without my phone
Your phone needs a web browser or an app. An app for every airline you ever use? You already have a web browser.
They could SMS but its more expensive to send, often even more so for customers on roaming to receive.
Nothing else is universal.
I think there are much better possible solutions. An open notification standard or reasonable pricing of bulk sending SMS would do it.
If browser notification permissions would have a TTL, I'd might considering it. But until this happens I won't allow anyone to send me browser notifications. And even then I'd be very picky.
And yet I’m sure airlines will push you towards the app every time!
Generally, the recommendation is that you get to the airport at least two hours before your flight departs. Ideally, you shouldn't be rushing to try to get your plane.
Granted, the world has changed since that was first a recommendation, but even in today's connected world, it's still a good idea to get there two hours before departure, in my experience.
A lot of delays are known much earlier than that. For example if a flight gets seriously delayed taking off and the plane is going to turn round and return, then the return flight will be delayed.
In any case, once at the airport delays will be announced and shown on screens. Once you get there you do not need phone notifications.