Sarah, sorry for being blunt but you really seem to focus a lot on things that are totally beside the point.

It's not about being appropriate or inappropriate. You as a client/patient can say whatever you want whenever you want. That's part of therapy and also assessment - by saying whatever you feel is relevant to say will give the therapist important information about you.

However, you seem to think that if you say something like you want the therapist to say something nice to you in the beginning or after the session then that should mean that the therapist will start doing that. It seems that you haven't really understood that the job of the therapist is not to please you but to help you start working on yourself. The fact that you give feedback does not mean that the therapist has to change their conduct or have to conform to your feedback. Rather, the feedback will give the information about the person you are and of course, sometimes (or perhaps even often) some changes are in order but this is something that the therapist decides based on their best judgement and knowledge about you and it's definitely not something you will dictate. Because if the therapists would submit to what you want then you could conveniently demand to avoid all these difficult and painful problems you have and it's the job of the therapist to make sure that you can not avoid them.

You have seen so many therapists and none of them has conformed to your standards. How convenient, isn't it? Because this means that you can dismiss them by saying that none of them was a good match and you can conveniently postpone looking at yourself and your problems.

However, maybe it would help to remind yourself that looking faults in therapists is not helping you in any way. You have a limited life time yourself and while you still have time right now to make a life you want for yourself, each time you are focusing on a therapist instead yourself, you are just wasting your own life and your own time. The time goes by fast and you one day you might find yourself in an embittered position where you realise how many chances you were given and how you turned them all down and how it's then all too late to change anything. But that's eventually all up to you.

My sincere and earnest suggestion for you would be to just stop focusing on therapists and their faults and try to focus on yourself and what these therapists offer to you. It's going to be very hard because you are so used to vigilantly observing others, but it's the only thing that can help you.

You have been several times offered psychoanalytic therapy for free by a public healthcare. This means that you have been basically offered for free the best therapy there is for the depth problems you have. You can turn it down and keep looking for a unicorn, and it's your choice. But it does not seem that it has helped you so far.