You may consider usink keys / hotkeys.
Selecting
F6 - acces your url for focused active tab
import pyautogui
pyautogui.click(0, 200) # a random click for focusing the browser
pyautogui.press('f6')
CTRL + TAB - next tab
pyautogui.hotkey('ctrl', 'tab')
CTRL + SHIFT + TAB - previous tab
pyautogui.hotkey('ctrl', 'shift', 'tab')
ALT + TAB - for another chrome window ( usefull if you know you are only chrome opened)
pyautogui.hotkey('alt', 'tab')
CTRL + T - open new tab
pyautogui.hotkey('ctrl', 't')
CTRL + W - close curent tab
pyautogui.hotkey('ctrl', 'w')
Copying
For copy the url, you can use either pyperclip...
pyautogui.hotkey('ctrl', 'c') # for copying the selected url
import pyperclip # pip install pyperclip required
url = pyperclip.paste()
print(url)
either clipboard module...
pyautogui.hotkey('ctrl', 'c') # for copying the selected url
import clipboard # pip install clipboard required
url = clipboard.paste()
print(url)
More about hotkeys into : documentation1, documentation2
PS: I have Chrome x78.0 on windows 64 bit os, and it's working for me. :)