Music notation is about problem solving: how to tell the musician what is going on? Specificall, what's going on musically (semantic) and what is going on technically (syntactic).
The 'real' reason for Klavar seems to be that it now includes a one-to-one correspondence from the piano layout to the music notation, like tabs do. Sounds good, right? But does music have a one-to-one correspondence to the piano layout? Except for C-major, the answer is no.
Klavar may offer a syntactic advantage, but it loses all semantic advantages of classical notation. In classical notation, a harmonic minor scale will look pretty much the same in F# as in Db. The key signature will take care of exactly what notes to play, but the practised musician will immediately recognise the semantics of what is going on. In Klavar, these scales will look wildly different, despite serving the same function in a piece of music. On the other hand, an accidental (a temporary change from the key signature) will be easily overlooked in Klavar, while it is clearly indicated in classical notation.
Classical notation has evolved to tell what is going on in the music, somewhat agnostic from the actual instruments involved. While it certainly has many drawbacks, it does convey the meaning of the actual music very well (even if it does mean that a beginner needs to count sharps and flats to figure it out).
Some other considerations
Pianos have only one key for each note. The most obvious problem, which hand plays what, is solved by having two staves. Klavar, in my opinion, makes this less easy by putting everything on one stave and showing which hand plays what with horizontal lines.
Guitar tabs solve a problem specific to guitars: how to show which string to play. This is important for various reasons. Not only is it easier for the guitarist to quickly see how to play, but more importantly, the same note on different strings will sound different. Most obvious is an open string versus a fretted string, but a high note on a low string will also sound different than the same note on a high string. This is a semantic advantage (what is the intended sound?), and thus has gained wider adaptation.
A final point is that the guitar has gained popularity as an easy and accessible instrument, enabling people to emulate their pop- and rock stars with minimal investment. Tabs are an excellent way for novices to play some of the songs without being bothered by a minimal understanding of the music theory involved. If pop- and rock music was dominated by raw electric violins and thumping cellos, I'm sure tabs would have widespread adaptation for bowed instruments rather guitars.