Temanor You don't have to agree, but this simply isn't how branding works. "Kagi shouldn't have to change" is the wrong way to look at this. It's not a question of justice. If it were, you're right, Kagi can use a "g" like anyone else. "The K is boring" is also the wrong way to look at it. It's not a question of style. If it were, you're right, the g logo has nice style. It is a question of branding. Picking the first letter of your competitor who dominates the market is, objectively, bad branding and especially so when it's not even the first letter of your product.
As long as most people in the marketplace associate a "g" in the context of search with Google (as demonstrated in my mini sample above), it's a measureably bad choice and counterproductive for a product with a fraction of a percent of the marketshare for search.
It'd be like launching a company with a swastika for the logo. It's a cool pattern, and could be beautifully designed, but good luck disassociating it with what you're already thinking right now without me saying it. And that brand has been out of circulation for almost 100 years. Google's empire is now.