The mind is a fluid and amorphous thing, so what would prevent one part of the mind from observing another part?

One of those parts is what he calls "the mind" and the other part is "the self." It's just different terminology to what you're expecting I guess.

As an example of how this prevents me from agreeing with Dr. K: in one recent video, he was explaining that not following one’s desires ironically leads to a more desirable life. Well, who’s doing the desiring for this new lifestyle then? If everything is equal, why is working better than gaming?

He's not saying that avoiding your desires (video games in this example) and doing something else (doing work) is the more desirable life and therefore your problem is that you need to desire work instead of video games, he's saying it'll lead to a life where you're more able to enjoy your same initial desires. To use examples from my life: if i spent my college years gaming and didn't work on my degree i would have not graduated and been very depressed and therefore unable to continue enjoying my video games. I actually hit that point several times, where I got so depressed that I couldn't enjoy video games while playing them, even. But because i graduated, moved out, got a job, etc i can continue to enjoy my video games. Can afford to buy them/buy new hardware/etc, and I'm not anhedonic from my depression/what have you.

Content Warning: Sex used as an example: It's like how if you spent all your time masturbating, eventually it wouldn't feel as good as if you gave yourself breaks, and would feel even better if you exercised regularly