The Langlands program is not similar to homotopy type theory. It is 'unifying' only in the sense that it makes striking, deep, yet very concrete[0] connections between 2 seemingly disparate areas of math, and because of this the tools needed to make progress in it end up touching all areas of mathematics.
Homotopy type theory is more for about foundations of math stuff, i.e. which axioms you use to prove things. ZFC is by far the most common system to work within, not HTT.
[0] ..despite taking years of study to understand the conjectures
Unfortunately the "geometry" involved is not the kind of geometry that lends itself towards being easily drawn to give understanding.
If you had to draw a picture of it, it would just be a sphere (since it shares many properties with a genus 0 complex curve).
But it's a very different beast than usual geometric shapes. A lot of modern number theory is motivated by an analogy between "p and t", where t is a parameter in a polynomial like t^2 - 2, and where p is a prime number. This space takes the analogy to an extreme where p literally is the parameter for the space[0], like t would be a parameter for the real line.
[0]Not exactly true - true for the "disk" cover of the FF curve.
I would be surprised if this gets a 3blue1brown episode - this branch of mathematics takes a lengthy time to wrap one's head around. 3blue1brown topics tend to show up in undergraduate math, whereas the Langlands program (and the Fargues-Fontaine curve in particular) would really only be studied by a subset of graduate students studying Number Theory.
