How do therapists feel about their patient skipping sessions? Does it annoy them?

Of course, when you are a private practitioner who’s livelihood depends on collecting fees for service it can be annoying, but shouldn’t. If you have a cancelation policy, the patient will have already agreed to pay for non-rescheduled or failed appointments.
Annoyance short-changes the therapy. I always recommend to new therapists to allow themselves to simply sit when a patient is late or fails to attend. I ask them to notice whatever is going through their minds about the patient and the missed appointment. In essence, I suggest free-associating to the failure. Often a surprising thought or insight presents itself that the therapist can use in the next scheduled session. Often, the therapist is able to approach the possibility that their behavior or attitude shaped by counter-transference may be at the root of the missed session and the patient, by failing, is communicating something that they have yet to be able to put into words
In essence, failed appointments are an opportunity for therapist and patient to delve into deeper layers of communication, and thus, change involving both.