Crawling the web is costly. I assume it's cheaper to use the results from someone else's crawling. I don't know what Kagi is using to argue that they should have access to Google's indexes, but I'd guess it's some form of anti trust.
Let me add more: crawling the web is costly for EVERYONE.
The more crawlers out there, the more useless traffic is being served by every single website on the internet.
In an ideal world, there would be a single authoritative index, just as we have with web domains, and all players would cooperate into building, maintaining and improving it, so websites would not need to be constantly hammered by thousands of crawlers everyday.
Yeah not that cheap. There's a few articles on HN now about small, independent websites being essentially DDOS'd by crawlers. Although, to be fair, mostly AI crawlers.