I actually really like the stock browser except for the not being able to close it
If you're not using your browser the fact that it's in memory doesn't mean it is "running." Think of your memory as a room with a bunch of apps sitting around in chairs. There is a limited number of chairs in the room. If all the chairs are occupied someone has to leave before someone else can come sit down. But until you reach that point, additional apps can come into the room and do their business. Most will then take a seat and remain quiet until the O/S tells them to leave.
Some apps have periodic business to conduct so they'll be ready for you the next time you need them. In the case of the stock browser the little "near me now" line on the Google search page tries to determine periodically where you are. If GPS is enabled, it uses it to do that. You can turn off GPS and it will just sit there quietly unless you fill up memory with lots of other apps.
If you like the stock browser, switch your home page to about:blank or to a web page that doesn't utilize GPS. Or use another browser. The stock browser is the only one that utilizes the "near me now" functionality on the Google search page.