Providing options, with a professional and courteous tone, to rectify the situation is different than apologizing. Polite expressions of options are necessary.
But apologizing? As in, "expressing sympathy"? IMO, no.
Humans do not get offended when their car doesn't start because it's out of gas and doesn't offer an apology. They still love their car, they will still continue to use it.
Humans do not get offended when they click to a cable channel that they aren't subscribed for and an expression of sympathy isn't displayed. They will continue watching television and subscribing to cable.
Humans do not get offended when the hold a hammer by the "business end" and try to drive a nail in with the handle, and an "I'm sowwy!" flag doesn't pop out of it. They will go on to learn the correct way to use it.
Computers are merely another inanimate object. Ultimately it's a system that serves the user when used correctly. A bank teller on the other hand, should apologize because there is the appearance of discretion in the bank teller's undesired response. Applications have no discretion in their actions. Humans are fine accepting that their actions led to an undesirable outcome, especially when avenues of correction are explained. Constant expressions of sympathy are annoying at best, and at worst can plant the seed of mistrust in the design and value of the system.
This doesn't hold when it's reasonable to hold the developers at fault for the undesired outcome. For example, if a feature was removed or redesigned without fair notice, an apology might be appropriate. In this case, they're failing to meet a certain expectation that had previously been set.
Update:
Upon further reflection, I recognize why I feel so strongly about this. I've worked on many public-facing applications before, from finance to social media. In each case, the use of apologetic or sympathetic expressions matched the site designer's personal belief in the value of the system and the correctness of the design. Where they had also expressed doubts about the success of the project, apologetic error messages were standard. Where they believed the project was well-designed and, if used correctly, would benefit the end-user, merely professional messages were used.
I now find myself inadvertently inferring such levels of confidence from all application messaging/copywriting.