I come at this issue from both directions. I am a writer who does not have a beta reader, but I beta read for someone else. I catch a lot of big and small things for the person I'm helping out. They have been very grateful for my assistance. I've beta'd for some other people too. One person cannot catch everything.
Yet I don't have a beta reader. Why? Because it is a lot of work for someone to do. I write what some people might say are long chapters and update often. Finding someone willing to put that time in is rough. Let alone someone able to do so when I need them to?
Having someone else re-read your work is never redundant. I assure you we both miss a lot. You know your story already. Some things will always be a blind spot because of this foreknowledge. I have also gone back to doublecheck something for plot and found an inexcusable grammar mistake which no commenter mentioned. It happens. Nothing is perfect. There is a reason a lot of professional authors say to leave a manuscript lying around for at least a month or so before you go over it again.
Except this is for fun, so I'm not too worried about it. No one is going to crucify me for one wonky sentence or some misplaced grammar. For the most part, fanfic readers don't say anything about your story at all (whether they enjoyed it or not).