What is worse: 10,000 agents running daily targeted queries on your site, or 1 query pulling 10,000 records to cache and post-process your content without unnecessarily burdening your service?
At the end it's all about the impact on the servers, and those can be optimized, but this does not seem to happen at the moment at large. So in that regard, centralizing usage and honouring the rules is a good step, and the rest are details to figure out on the way.
If (for instance) my content changes often and I always want people to see an up-to-date version, the second option is clearly worse for me!
My apprehension is not with AI agents per se, it is the current, and likely future implementation: AI vendors selling the search and re-publication of other parties' content. In this relationship, neither option is great: either these providers are hammering your site on behalf of their subscribers' individual queries, or they are scraping and caching it, and reselling potentially stale information about you.