Or at least have maybe the ability to set some other answers as assisted answer. I find myself finding an acceptable answer unusable while other answers underneath it might be more suitable. Would it be safe to say that the accepted answer should only make sense to the asker whether or not the question is important to others?
Accept the best answer and upvote any good answers. I don't see any reason for this to change, other than trying to spread reputation around. If you have three accepted answers, it sort of defeats the purpose. The green checkmark should be used by the OP to reward the answerer who helped him/her the most, not to decree correctness, since often the OP is about the worst person to determine who is correct and who is not. That's what votes and comments are for.
If you have an answer that completely answers your question and it's the "best" of many such, then I believe that you should mark it as accepted to signal to the rest of the world that this answer is the one that solved your problem in the best way. If you have several answers of similar quality, but none of them completely answers your questions, you can either choose the best of them -- and comment on what additions are needed, update your question hoping to get a better answer, or, if you can put several answers together to get a single, best, complete answer, write your own answer combining all of the best parts of the incomplete answers into one and accept it after the waiting period.
The goal is, I think, to get a complete, accepted answer to every possible question, not to push points around on the table. You'd certainly want to upvote all the helpful answers whether you accept them or not. And I wouldn't create my own answer unless none of the other answers is acceptable.
In this case I would upvote all answers you would like to accept and compose a combined answer yourself and accept it. Give credit to the answers you used for your answer. Of course that way nobody gets the 15 points, but this site is about great and helpful answers in the first place ;).
This holds true if all answers contributed to the solution of the problem somewhat equally. As this is a community driven site with no ownership, you could also accept the most helpful answer and - provided you have neough reputation to have edit power - edit the others in. Clearly mark and attribute them, and again upvote the other helpful answers.
Look at this duplicate as well.
To offer a case in point:
Here is a question with multiple equally satisfying - shall I say perfect? - answers: https://aviation.stackexchange.com/a/57758
I think in such (rare) cases, it should be allowed to mark >1 answer as correct.
What about once per week ?
Or twice a month ?
This would both assure that:
- in normal cases (i.e. the vast majority of questions) the best answer is marked as correct
- in rare cases where 2 perfect but different answers exist, both can be awarded