I think, error messages should put way more focus on what can be done to make it work - rather than on what went wrong.

The user wants to achieve something, and the general job of the software is to help the user achieve their goal.

If the goal of the user is to book a flight - then the software throwing an error about the request form not being filled out correctly, basically constitutes a bureaucratic hurdle.

"You will not be allowed to book your flight, unless you ask for it in exactly the right way!"

Sadly, software often has to be bureaucratic about it's input - but that's not actually helpful to the user. So you could argue, when the software rejects the users request because of not meeting bureaucratic demands - the software is failing it's primary job: to help the user.

A perfect software would not have to show an error message ever - because it would be so good at guiding the user, that everyone would intuitively know how to do it correctly. But "Perfect" does not exist.

So any software will at some point inevitably fail to guide the user to their desired outcome and be forced to display an error message.

There's nothing inherently wrong with being apologetic in tone for failing to guide the user and putting such bureaucratic stumbling blocks in their way.

But the error message should be focused on helping the user. On making things work. It should not be condescending nor assigning blame. It should also be short and concise, and respect the user's time.

And that means that there's no space for polite verbiage and apologetic phrases.

So, apologetic tone is fine, if it can be done without increasing the length of the message.