Excluding Chrome, Firefox and Safari means that you are now relying on some random developer to understand security and privacy and as a software developer for over 40 years I can tell you that this is a fools errand.
Don't get me wrong, the big three absolutely have privacy issues, but they can be mitigated in many different ways without compromising on security.
For example, you can force DNS requests to one of your choosing, you can run them in incognito mode, refuse cookies, run them inside user accounts without personal information, etc.
I tend to run individual instances of a browser in incognito mode and am very conscious of which tabs are open in which instance, so websites cannot steal information from other tabs.