Try installing a new browser from the Play Store, then follow a link from some app (forum browser, email, anything). Since you've installed a new browser it should ask you again which app you would like to handle the intent, in which case you can choose Chrome and tell it to make this the default.
I used to remove the stock browser on my previous phone and it worked fine until I installed one app which needed to open a browser page for one task and clearly had been coded to use the stock system browser, which I didn't have installed! Apart from that one app (which you are unlikely to be using yourself, as it has a pretty specialised audience
) I got by fine without a stock browser. Keep a backup if you go that way though, and maybe consider "freeze" rather than "uninstall".