Hi, wow, so let's keep calm and see why the problem in the video (if it exists) has nothing to do with the OP asking which formats to use for import and why there are jaggy and white edges. Which has largely been solved, I take it ? Any way, rough edges probably have to do with the (lack of) resolution.
A screenshot and the offending image for download would make it easier to give a qualified answer, if it is the source image or somewhere in Godot. As said already, there should not be problems with PNGs.
To the video: the guy shows a linear expression of ax + by + c*z with x,y,z from 0.0 to 1.0 such that their sum is 1.0 and a,b,c each from 0 to 255. All values floating point (conversion else no calculation on a computer). That alone is a hint to a programmer that checks must be made, simplistically said (irl it maybe more complex) clamp the sum of x,y,z to 0.0..1.0, check ranges of a,b,c if they are 0..255, or simply check if the expression results in 0..255 and clamp/expand/compress if necessary. Specifically when things come from an external source. Not doing this is careless and inviting disaster, likely to get garbage values, maybe from precision problems alone. Still, these should not make a phone unusable, if they do, there is really bad software involved and possibly more serious problems looming than just a blocked phone (speculation mine).
Anyway, when solving the expression and the outcome is > 255.0, then there is not yet an error (as claimed by the video guy). That can happen and must be accounted for as said, in any way that leads to a safe conversion to the narrower range of 0..255. Or, if a repair would result in too much distortion or false colours or whatnot, an error must be raised. This is really basic stuff for a coder, and should not break any functionality. And that's (among other things) why I said the video is clickbait and does not really explain anything.
This is all only valid if we actually accept the video to be genuine, of which I have expressed my doubts 😜
Edit: the same arguments are valid for the generation of the image by some software, of course.