Does NASA still need SpaceX? What about other space agencies?

Yes it does, and the article says why and how. Other space agencies... that's a very different question. Only China can be comparable to NASA in terms of breadth of space initiatives, and China has their launch industry. Everybody else mostly doesn't use as many launches per year as SpaceX is capable of doing.

I guess the answer depends on whether you see NASA's goal as "to do cost-effective science" or "to do cost-ineffective rocketry" (since the express purpose of e.g. the SLS is to appeal to congress by providing local jobs as economy stimulus)