Do you have another means to settle in the UK (for example, is your wife a citizen of the UK or another EU country)?
This is a lot of speculation based on my own understanding.
If you are hoping that getting a degree in psychology will provide an opportunity to live and work in the UK, you may get lucky and it will work out, or it may not (as your friend found with their English degree).
My personal experience with the counselling field is that people are usually self-employed/independent and don't work for a firm or company. If that is the case, I don't think you'll have much luck with being able to stay in the UK after your degree, as an entrepreneur visa is extremely difficult to obtain. If you are able to find employment with a company/firm, they would need to sponsor you and pay you a minimum specified salary based on the UKBA's guidelines. If you obtained a degree in the UK, the sponsor would NOT need to prove that no one else in the UK or EU could do the job. However, if you looked at moving to the UK after you obtained a degree in the US, the sponsor WOULD need to prove that no one else in the UK or EU can fill the role.
I think that most people find that obtaining education in the country they will obtain work in is the most appropriate path. The US and UK are not great at recognizing a qualification other than their own.
Yes, school in the UK (or university as you will quickly learn to refer to it as), would be a great way to experience life in the UK and see if it is for you. Many people love the UK for a holiday and don't take well to actually living here. Of course, many do take well to living here but it's not always an easy transition.