Careful!

Make note of which 70-200mm f/4 lens version is being discussed. There are three distinct versions:

1. Original poster was discussing the newest and current EF 70-200mm f/4L IS USM "II" (2018).

2. The "Five Minute Fix" on Youtube is being done on the earlier EF 70-200mm f/4L IS USM (2006). This version was discontinued when the "II" was introduced. It's also very important to note that this "fix" makes the lens completely unusable in auto focus! After doing this it will be manual focus only. Accidentally using AF mode with the "fixed" lens might do serious damage to lens or camera or both. The guy in one of those videos goes out of his way with disclaimers!

3. There's a longer (45 minute) fix shown in a different Youtube video, involving a lot more extensive teardown of the lens. But that's being done on the EF 70-200mm f/4L USM "non-IS" (1999), the oldest version of the lens, although it's still in production and being sold today.

Personally I've never had any problem with the 70-200mm f/4 IS USM that I've used for several years and bought used. But I also pretty much exclusively use it in AF mode and it appears most problems are related to using manual focus. I do sometimes use de-focus/re-focus technique with it... as I do with most of my lenses. That involves using the manual focus ring to deliberately de-focus the lens, to quickly cause AF to re-focus. Issues with the lens' manual focus mode might be more of a concern for someone using the lens for video, which may use manual focus a lot. I use it for still photos only. However, there are some online users who reported eventually seeing problems with the AF too, some time after first seeing the MF issue.

The design and function of the Canon 100-400mm lenses isn't very relevant. The 100-400mm II's "tensioning" ring has nothing to do with focusing, which is what's being discussed above. That tension ring adjusts the drag on only the zoom ring and mechanism, and at the one extreme can be used to lock the lens in it's most fully retracted (100mm) position for storage. The original 100-400mm "push/pull" zoom has a similar locking mechanism.

Also, both the 100-400mm Canon lenses are not internal zooming... They both grow quite a bit longer when zoomed to their 400mm setting. They are internal focusing, don't change length when focus is adjusted. A significantly different design, all the Canon 70-200mm lenses are internal focusing and internal zooming. They never change physical length when focus or zoom settings are adjusted. Generally this type of lens doesn't need or use a zoom tension/locking mechanism. That's usually only needed on lenses that are not internal zooming. For example, it can help prevent the zoom self-extending while carrying it... sometimes called "zoom creep".

All this has me worried about my own 70-200mm f/4 IS now! I'm resisting temptation to get it out and test the manual focusing, to see if it slips as described. It really shouldn't matter to me, since I use the lens pretty much exclusively with AF. Maybe I'm better off not knowing if MF is slipping!

No, the difference is the electronics, not the optics that are the same in the vII and vIII. The weakness of an 1.4x extender is that on a shorter lens, this 40% increase moves just from 200 to 280mm which really isn't much of an increase at all. Yes, then there's the crop factor. If in the stands for a HS soccer game, these tools combined can help give a clear image of players at midfield, a usage of this type of configuration I've used in the past (with more cropping). But, the reality is that the 1.4x extender just doesn't add that much, the loss of 1-stop of light being somewhat more impact than the increase in the effective focal length.

Since the Mk III works with your lens, I assume that the Mk II will also, but, to be absolutely sure you should check.
You will see some degradation of IQ, but, with a lens that sharp it should be minor. The auto-focus may slow down a slight bit but I doubt it will be noticeable. I have a 1.4x III that I used on my EF 70-200 f/2.8 L II with both a 7DII and a 5DIV several time with very good results. I only noted slight differences in IQ and auto-focus performance. What camera body will you be using it on?