uBlock Origin does actually block network requests before they leave your browser. The main difference is that Pi-hole can block requests from any application on all your networked devices. Theoretically you can achieve the same effect of a Pi-hole with a hosts file (except DNS caching), but that would involve setting up and maintaining one on every device you own… if that’s even possible (looking at you, Android and IOS)
If the breached data contains your account passwords, cryptography keys, credit card info, etc you should update and invalidate that information. Once that data is out there, there isn’t really anything else you can do but make that data no longer useful.