I'm fine with the government not running the details of their business. But the airlines took a bailout during the pandemic and used some legal, but grey areas, to adhere to the rules while going against the ideas that established those rules. Here is a decent write up detailing some of that.
https://www.mercatus.org/research/policy-briefs/2020-bailouts-left-airlines-economy-and-federal-budget-worse-shape#:~:text=How Big Were the Bailouts,billion%2C and %2414 billion).
Even if there is no legal means for DoT to reclaim or force them to pay back some of the loans they got which will otherwise be forgiven, government should have a way to deal with big business. Maybe after the first time all the airlines had trouble over the summer, instead of writing a letter asking them to get better, maybe advocate for an updated Passengers Bill of Rights which would put more financial incentive on providing reliable service. Southwest specifically had
Additionally, even if this specific issue is something that Buttigieg had zero ability to proactively deal with, it wouldn't be the first time an issue has blame unjustly put on someone and opens up some other legitimate issues. I wonder if that's what we'll see here.