Or you could just sell shares to the public, since you're a public company. That would get more people involved in the market which is a good thing. If the government were to buy the airlines to inject cash they ought to assign the shares to individual americans.
If they were publicly owned, the prices of flights would just go down until there was no profit. Also, the CEOs wouldn't be paid tens of millions of dollars per year [2], so again, prices would go down.
[1] https://www.cnbc.com/2019/06/02/airline-industry-cuts-profit...
[2] https://skift.com/2019/05/29/united-wont-pay-munoz-a-full-bo...
And even if we were to make airline public, wed just be reversing the trend over the last few decades. There is a reason why so many nationalized airline went private, the government kind sucks at running businesses (not always, but often).