Ad blockers, pop-ups and the player window
Two browser settings decide how a client behaves, and only one of them stops it working.
Pop-ups have to be allowed. The client plays in a separate window. If your browser blocks that window there is no client at all — allow pop-ups for this site and press start again.
An ad blocker is fine, and you do not need one. AdBlock, uBlock Origin or any of the rest: fewer adverts in front of a video is a shorter wait for the same credits, so having one costs you nothing and lacking one costs you nothing either. We do not ask anybody to install one.
What does hold a view up is anything drawn over the player — a cookie dialog, a consent notice, an extension's own prompt. The video stops behind it, and only you can answer it. Answer it once in that window and the rest of the session runs clean.
The finished player windows stay open, and there is no setting that closes them. Every session keeps one window open — the newest one it opened — and a browser client cannot take the older ones away once YouTube has them. If you are running several sessions you have several live windows, one per session, each in front of its own pile. Closing a finished one interrupts nothing, so close them as you go; close a live one by mistake and the session simply says so, and one press starts it on the next video.
A window you leave open does not sit idle. It plays to the end and then YouTube's own autoplay moves it to something else, and it keeps going. That is worth closing for its own sake, whatever it is doing to the speed of your machine. The desktop app reuses one window instead, so nothing accumulates.
The client pauses after a number of windows and asks you to close some. Left running all night the windows add up until the browser — and sometimes the computer — stops coping, so the client stops first and says so on the running list, which also shows how many it has opened. Close any that are still open and press "Watch the next video"; nothing is lost while it waits, the sessions stay open and the count starts again.