That depends on the therapist and all the other variables (how things ended, length of treatment, etc.).
The notion that the therapist “failed” the client suggests something that is binary and one sided. Therapy is a collaboration between therapist and client. As a therapist I do my best to facilitate patients growth and positive change, but at the end of the day, it’s patients that have to do the work and make the changes.
It’s not uncommon for patients to think the therapist is supposed to “fix” them. Doctors may be able to fix a broken bones, but therapists cannot fix an individual’s mental state. They may be able to provide them with helpful tools, help them see things that they had not recognized on their own, and provide ab environment in which they can identify ways to make positive changes, but this only happens with a lot of independent work by the individual.
Therapy is not done to someone, it happens with them.
I understand there are many situations in which patients may feel therapists hand not done a good job. Once therapy is terminated they’ve lost the opportunity to get the therapists help in understanding what is happening, or what is going wrong, in the treatment. Sending a letter like this after termination sounds more like an attempt to lash out at the therapist and hurt them, then to really gain any insight into what happened.
I’m not saying that therapist always do the greatest jobs. They don’t. They’re flawed. Some of them may be incompetent. You need to ask yourself what is to be gained by writing this kind of letter.