Safari does not change focus to the web page after address bar search
I was searching for the same answer that you are looking for, and I know that this answer is not the exact solution you were looking for, but it may ease the frustration a bit until a better solution is found.
I remapped my Caps Lock key to function as the Esc key, so after I do a search from the address bar, hitting Esc is a quick and ergonomic keypress away that lets me keep my fingers on the home row.
If you're extra fancy and like to remap Caps Lock to Control, there's a solution to that as well. Using the free app called Karabiner Elements, you can remap the Caps Lock key to both Control, if held and another key is pressed, and also Esc, if pressed by itself.
If using Karabiner Elements, and you need Caps Lock functionality, you can either assign it to the Control key or, what I do, is assign it to toggle when I press both left and right shift at the same time.
I hope someone else finds a true solution or that Apple updates Safari with at least an option to change this behavior.