uBlock Origin offers two element-picker modes. The lightning-bolt tool deletes whatever you click on, adding a rule blocking that element for future visits. The eyedropper tool does the same, with a preview option and more control over the selector you filter out; this is useful with modal overlays, where the prompt itself may be nested in a screen-filling div, blocking the underlying content.

I use these features liberally - for cookie notices, "subscribe to our newsletter" nag boxes, sticky bottom banners, chatbot requests, and everything else which gets in the way.

In addition to per-site rules, I have a blanket block on OneTrust, which seems to be the most popular cookie nag system:

  *###onetrust-consent-sdk