2 reasons:
1.) a buy in $ gives you access immediately to a fixed % of the corporate accounts receivable. Without any sort of buy in, the shareholders would essentially lose a good chunk of their AR as the time on AR is so epic for medical collections vs. other industries.
2.) a buy in gives you immediate access to an established contract i.e. good will of a business. this is huge and can be very very valuable depending on the natures of the contracting entities. As hard as it is to imagine from an accounting POV, some contracts are so "evergreen" they never are removed from the primary holders involuntarily unless they commit some crime.
No 2 reason requires ALOT of research. Groups that recently have taken contracts from crappy revolving door hospitals or medical groups are worth far far less in terms of the good will than say a group which has had the contract for 50 years. You need to understand exactly what you are buying.
No 1 is fairly easy to discern IF you are an intelligent consumer of financial accounting, if not you will need an expert to help you.
To directly answer your ?s:
1.) do you get any money back from a buy in if a group loses a contract? No. The only exception I can think of is where the group dissolves almost immediately after getting a new partner, which I have seen and is a massive legal mess.
2.) do you get more money if business has increased at the time of a dissolution? Only in the sense your share of the EBITA will be greater.
Buy ins are not predominantly for assets in a business, unless your business is (surprise surprise) asset driven. Service oriented businesses are always driven by non-asset considerations first and foremost, even for large law firms as they tend to put things like RE in separate LLC structures completely.
Hope this helps.