That's not how it works, though.
First, if it is not possible to blind the patient to the intervention and/or provide a meaningful placebo (which also happens with surgery and drugs with prominent side-effects) that's not a big deal, it simply changes the interpretation from "the effect of a particular regime of psychotherapy as such" to "the effect of a particular regime of psychotherapy, including expectation effects". Those expectation effects also exist in real life, if you go to a therapist you really really hope you'll get better and may be motivated to make a few changes in your life independently of the treatment, so if anything the trial becomes more realistic. Note that you'd still use a control arm so that if people get better over time without treatment, you're not exaggerating the effects of the intervention.
Second, almost every randomized controlled trial compares an experimental group against a control group, you are assigned to a group and don't move between them, and what that means is that we can only compare whether the treatment works on average, not whether it worked for one person in particular because, as you say, we don't know what would have happened (counterfactually) if that one specific person had been randomly assigned to the other group. But again, this hardly matters, because we can still know how well a treatment works on average. What you refer to, an experiment in which every subject receives every treatment, is known as a crossover design and while that design has its advantages, it's not that common because it is harder to pull off and while it can be more efficient (need fewer patients to attain an estimate with a given precision) it is not "less biased" or inherently more trustworthy.