"Would you like to drink some coffee or tea?"
First let me correct your question. It should be "爲什麽不用~." The subject "we" or "people" referring to an unspecified group of people is redundant in Chinese and we often omit it. The answer to your question is, yes, you can use 想, like "你想要喝" or "你想喝." When meaning "want," 想要 and 想 are completely interchangeable.
You can test if a question should end with 吗？ if the answer can be just "Yes" or "No".
For this question, you need to answer it with either Coffee or Tea; you cannot just say Yes or No.
还是 may be used in different constructions too, but in this one it indicates a question, so you don't need 吗. The construction also implies that you may only chose one of the options (you can't have both coffee and tea).