A therapist typically isn't going to refuse to offer their professional skills to a patient/perspective patient. Or just drop someone on a dime. There's always reasons. Most are usually glaring, to the therapist. Maybe even the patient, but not always.
The most common reason that a therapist/psychologist is going to refuse their services is that the person is an unreliable narrator. Due to problems with accuracy, credibility, maturity and a persons mental state. Other common reason is that the person isn't open to receiving help/advice, or is a liar. Or is defensive/combative/argumentative. Another common example, is similar to not clicking with a CFI during training, you and your therapist also might not click. And one or both people decide it best to end the relationship. These types of behaviors are counter productive to treatment, so its just better to cut bait, often it isn't personal. But yes liability does play a big part in those decisions.
As a personal example, my younger sister recently informed me a few months ago that her last two therapist ended their professional relationships with her. She feels that she has childhood trauma in the form of neglect. My mom was/is a good person. But she loved and valued education. With her five degrees she helped provide a very good life growing up for our family. But my sister feels that with my mom getting her second Masters and her choice afterwards to get her Ph.d, that she was neglected. She feels growing up that she didn't get the same attention that my two older siblings and I did. My mom even feels the same and has a lot of guilt about it and has said its her biggest regret. My sister explained that she was told by her therapist, that she was closed off and defensive. She said both therapist ended up asking her what was her motivation in seeking help, if she wasn't willing to put in the work and was always closed down to any and all suggestions? She says she's done with counseling. Her choice based upon her behaviors.
Not all therapist/psychologist are well versed on every issue. For example my mom though she never practiced got her Master in Psychology and went on to get a certificate specializing in treating children with social and developmental disorders. Because she was teaching Special Education for years dealing with children with learning disabilities, neurodiviergent disorders and developmentally delayed kids. Trauma's and behaviors weren't her bag. My stepfather was a Psy.D. When he was alive he owned his own counseling business. He didn't like the leaving money on the table as he used to often say. So he got multiple certificates (like an add-on in aviation) to be able to be well versed in a multitude of treatments so he wouldn't have to send a perspective patient out of his office on a referral to someone else.
If he had a specialty it was treating kids, forensics and trauma related disorders and behaviors. He always said that his favorite patients to have were borderline, sociopaths/psychopaths and narcissist. He explained that they tend to like counseling, because they love the attention aspect of it and that they love to talk about themselves and hear their own voices. He always demanded six months cash up front from them though, because they could usually afford it. As they tend to have short attention spans and typically don't finish counseling. But he didn't care because he already had their money. Neurodivergent patients weren't his bag, it was more my mothers. Likewise she didn't deal with the Cluster B & C types at least in a therapist setting. Most therapist, psychologist and Social Workers, (who if they have their MSW (Masters of Social Work and an LCSW cert. Licensed Clinic Social Worker are therapist.) on their websites are going to list their qualifications and specialities. So, if you're neurodivergent it makes no sense to go to a clincian that specializes in say early childhood development disorders. Also therapist, Social Workers and psychologist can't prescribe meds. Most psychiatrist or nurse practitioners with a mental health Master's or doctorate degree only prescibe meds and don't offer therapy. Because they're medical professionals and not therapist/psychologist. I'd seek out a therapist/psychologist, well versed in congitive behavioral therapy (CBT). Or the new fangled Dialectic behavioral therapy (DBT). Both have their foundation in talk therapy.
Hope this helps some.