You can remove the Google search bar and replace it with the one of your choice.

You can sync KeePass files automatically between devices using plugins. Takes time to set up at first but afterwards you have the best of both worlds completely for free.

I have been using the same setup for the last 4 years. Took a while to set up but there was a step by step tutorial in a computer magazine that I followed. From then on, each time I logged into an account I changed the password and put it in KeePass. Now I have about 350 passwords stored. For me it was important to have control on the password DB and not be dependent on a company.