I would have privacy concerns, because I'm not using Google all the time for my searches and there is no need to give them even more data.
The only solution I'd accept is one where all data is stored and indexed locally and there are good guarantees that deleted entries are actually deleted and/or wiped from the indices.
I’m not suggesting that you have to use google for the originating search. But you would need to use CSE to index any content you wanted to search for later. Arguably you could do this with a new google account if you’re concerned.
A few years ago I would have been excited about this, but I personally won't be giving Google any more data unless I'm forced to. I'd love a self-hosted and local-first option, but either way I wish you good luck on your potential project.
The only solution I'd accept is one where all data is stored and indexed locally and there are good guarantees that deleted entries are actually deleted and/or wiped from the indices.