Should I explain how I identified a relevant supervisor in my SOP? I spent time reading a professor's papers, and I developed an idea about how to expand the professor's work. I told the professor this and got the professor's approval. Now the next job is to persuade the admission committee (adcom).
Should I tell the adcom about how I found the professor while writing my SOP (statement of purpose)? I don't see a problem with it. I would be happy if a stranger told me how they found me. So I wrote about it the first time I contacted the professor. However, the adcom is not the professor, and I'm not sure if they care or not. I think that they only care about the benefit that I will bring to the department, as well as the approval of the professor. How I reach them is not important.
But I don't have any reason to remove that part, right? At best, it will show how hard I worked to reach them. At worst, there will be nothing to lose.
My draft is as follows:

I have read some of the most recent and commonly cited papers of Professor X, starting with Paper 1 written in 2014. Among the introduced methods, I found the Method the most interesting prompting further study. I then focused on Paper 2, written with Y. Its introduction on Quanta Magazine gave me a great insight into the Method. However, there are known limitations that Prof X is keen to discern. For example, when the system undergoes rapid changes the method breaks down, the reason for this remains unclear.
My hypothesis is that the changing of the system might suggest that... (this paragraph is for showing my idea to expand the work)

A minor question: should I include this in the email to professors too? Why or why not?
I don't know if adding this information is a good idea, but I think your current text is not.
In the draft, as it stands, you are just pointing out highlights, that proofs that you have skimmed through two papers. You are not showing any hard work because you don't even need to understand the papers to say that the method is interesting and to quote limitations they themselves described. I am not saying you haven't worked hard, or that you don't understand the paper, but just that it doesn't show all that. So, it risks backfiring at you because it sounds sort of pretentious.

If you want to show deep understanding, you would need to "cook" some new knowledge, like relating several papers, or relating their work to other people's. Mind that it will be read by people not familiar with the paper, and they may not know what is your own work, and what is just regurgitating.

One tip I can give you is to try to give it a focus from some other field you are an expert in. For example, sketch out how you would like to introduce a different mathematical modelling technique (like, instead of solving a single differential equation, you model your system as a network of interacting differential equations, where the topology comes from blablabla). This will give them a reason to hire you specifically: you are not just another mini Prof. X, but someone that would be bringing in new knowledge to the group.

Just make sure you run this through your local expert, to make sure you aren't saying anything silly, or that has actually been done before.

If after doing this you believe you are adding new useful information that would make you look better in the face of the admission committee, include it; otherwise, don't.