The search engine experience does feel “dated” to me. But I hear you on the claustrophobia you feel when you are locked into a Plugin/“AI App” experience. But think about the consumer perspective …
What I think the motivation of Plugins or “Apps” interacting with AI is that it extends the AI to such a higher level, and it’s the level folks (consumers) expect “True AI” to look like. They want the AI to know of information that is current, not just before the information cliff in 2021. They want it to book flights, recommend restaurants, give recipes and order ingredients for you, etc.
Sadly, the API has now become 2nd place to the direct-to-consumer buzz that ChatGPT and Plugins provide. They are simply capitalizing on the excitement.
But there is another reasoning as to why, I think, this is happening. Mainly, that any one of us devs could have created ChatGPT by riding off the current API. But the reason why this hasn’t happened en masse is that the roadmap of the API is opaque (you can only guess based on the past trajectory) to now non-existent because ChatGPT has disrupted everything. And how could I make money without cutting out the middleman (which would be me). So bottom-dollar, low margin solution is to go direct-to-consumer. This, of course, would all be flipped upside-down if their API was cheaper.
So there is a bifurcation, where you have ChatGPT/Plugins on one end, and general API use on the other.
ChatGPT is a direct-to-consumer cash cow, but possibly maddening for OpenAI since they aren’t historically a consumer facing company. And OpenAI has the API ready for direct consumption by developers, and as whiney as devs get, we aren’t nearly as bad as consumers. 
So, now, $$$$$$ (ChatGPT, but deal with consumers), or, past, $$ (Devs, and be happy not dealing with consumers). I think what changed the OpenAI culture was the investment/partnership with Microsoft.