I like that you are into musical topics, Juran!:) A capo is a simple manual device that basically does what you are aiming for. Most guitar players use it quite often. The guy in the cover image is using it. However, the app that you proposed would have some benefits over manual capo. On the other hand, the capo also has some benefits over using an app instead. So I'll talk about pros and cons (from my point of view) of both below.
If you start playing on a "zero" (basic) fret, with a capo you can only raise the key up, you can't bring it down, since guitar is simply made that way. If you are playing a song composed of, let's say D, Bm, and G chords on a zero fret, but want to transpose them down to be able to sing lower, an option is to count the distance between chords and simply play different chords accordingly on the same 0 fret. So if you want to sing a tone lower, you can play C, Am, and F instead.
An app would let you do this without counting or knowing the distances between chords, you could simply look up and play. However, the problem here is that if you are playing on a zero fret the options of bringing chords down are already very limited, if there's an E chord in your song, you can't bring it any lower. I actually had to think a bit just to give an example of the combination of chords played on a basic fret which you can still conveniently bring down (and that's by one tone only).
You can transpose chords to be able to sing lower with a capo too - chords played on higher frets might let you adapt your voice by lowering it slightly and the song would still sound nice, but you would be singing lower while playing higher, so that's technically cheating.
You could use barre chords instead of capo and in some songs with specific chord combinations a capo would limit your possibilities of playing wider range of chords (e.g. a chord on an 8th fret and a chord on a 2nd fret in the same song). So the app could simply show you a transposed song in barre chords and you'd still have the freedom to play lower than the specific fixed key, which is not possible while using a capo. But for beginners using a capo is really the easiest way. Barre chords are the most difficult to press since you have to press down all the strings with one finger, so the wider range can be sacrificed for the ease of playing.
An app adapting the chords to your voice automatically, as you suggest, would be another advantage over a capo. But finding the right key for your voice with the help of a capo is also very easy - you simply slide it up fret by fret and try to sing, such experimentation is also playful and joyful, you often discover a few keys that are convenient for you, instead of one. If the app would do it automatically, you'd lose that discovery part.
Finally, using a capo might just be simpler, it's often tiring to look at apps while playing (similarly how you have to look at the lyrics or chords on the screen when you don't know them, then keep touching your phone from time to time in order for the screen not to turn black, or additionally adjust the display settings, etc.). A capo you can simply put on a certain fret (experimenting part is easy as I said and later you get the sense of where to put it in advance) and play without looking at the screen, interacting with your phone, etc. It's also nicer to use manual things while playing instead of digital ones similarly to how dj'ing on an actual mixer with keys and buttons feels and looks way better than using only a laptop.