I think what is satisfying lies in the eye of the beholder. Though two versions of what a beholder might find satisfying are exemplified in this answer:
The first is the beauty of simplicity. Like without any (specialized) prior knowledge like geometrical series, graph theories, combinatorics and so on, you're able to visually see, how the proof works and that it works. So from the standpoint of a teacher or anybody trying to exemplify and explain that relation to someone else that's absolutely beautiful.
The thing is, that is practically equivalent to those:
But you could get away without knowing any of the mathematical notation or other levels of prior knowledge and abstraction, yet you wouldn't have to cut on the essentials of the proof itself.
The problem, or rather the beauty, of mathematical and complete induction is that it is not problem specific, but can be generalized to a whole range of relations. The idea is essentially that under the assumption that the relation holds for n (or even for all elements m ∈ N, where m >=0) then it also holds for n+1. So what's left is to prove that it works at all, by additionally confirming a base case, usually the start of the natural numbers by 0 or 1.
Yet as said if you follow this algorithm you'd have proven that it works, but you wouldn't have needed to understand what it does it in the first place, how people came up with that relation, how and where you could apply it or why you wanted to prove that in the first place. It's something like being a car mechanic without a drivers license that has never driven a car in their entire life. You know what you need to do to fix it, but you would only know in the abstract what "fixed" even means.
Now again eye of the beholder, for some the journey is what is satisfying rather than the destination, so that could be a nice proof, but practically you'd need prior knowledge of what it does and if you don't have that it makes you feel like having applied something without fully understanding what it does (and you don't have to, in order to apply the algorithm) and now having your fingers crossed hoping that it will hold.
And the other source of satisfaction is the second proof in that initial post, that is a proof that you came up with yourself, or that put your prior knowledge to good use, that was challenging and complicated but in the end all parts fell in place perfectly. There's just a beauty in seeing something working as expected, acquiring a complicated skill, being able to use your knowledge and the like.
But again that leaves you with the unsatisfying answer from the beginning, what is or isn't satisfying lies in the eye of the beholder and what scares of some might be the source of satisfaction for someone else and what has a simplistic beauty to some might be "too obvious" for others. Also some just care for the fact that it works, while others like it when they understand how it works and how they can make use of that.