If you put a video of another YouTube user in your playlist, let's say episodes of an anime, then make it public, will it get a higher chance of being deleted by YouTube? The anime episodes I put in my public playlist all got deleted (each and everyone uploaded by different users).
I don't think anyone outside Google can answer that with 100% confidence, nor do I think that they will disclose these matters. Keep in mind that this answer is mostly speculation.
There could be a chance that actions such as adding it in playlists etc. would "bump" the video in such a way that the algorithm/human in charge of monitoring copyright infringements would notice.
YouTube doesn't check or perform any complex operations on videos at the time of upload (probably to save on computing), but will take a closer look based on a set of predefined criteria. One example of them is the play count, as we can see explained in this video. We can draw an analogy that YouTube will not check copyright problems unless they see sufficient "activity" in those videos. It can be possible that actions like increase in play counts or putting them in a playlist would add to the "activity".
Also, by putting them in a same playlist, you would effectively be informing YouTube that those videos are related to each other -- if they find out that there's a problem in one, YouTube would probably check the others too.
YouTube carries a simple but yet effective algorithm. If any two duplicate content is found by their crawlers or bots, the effective action is taken. If the video is flagged for copyright infringement it will be taken down really fast.
Let's say if a video "X" is uploaded by a verified YouTube account and the same video on different channel which is obviously not verified. It will going to be pulled out soon only from the unverified account.
The simple rule of YouTube is that the platform is meant for users and the videos created by themselves. If you wish to upload those anime episodes I would recommend uploading it to your Google Drive Account.