Here is how you build just the current (formatted) date. See MacScripter / Dates & Times in AppleScripts.
set {year:y, month:m, day:d} to (current date)
# pad the day and month if single digit
set day_str to text -1 thru -2 of ("00" & d)
set mon_str to text -1 thru -2 of ("00" & (m * 1))
# make ISO8601 date string without time
set theDate to (y & "-" & mon_str & "-" & day_str) as string
The variable theDate now appears as "2022-01-08".
You can assign a keyboard shortcut to the Quick Action via System Preferences > Keyboard > Shortcuts > Services panel. Scroll down until you encounter your Quick Action (service) name, and on the right side, you will see a faint gray none. Click on that and it will change to an Add Shortcut button. Click that again, and enter the non-conflicting keyboard shortcut, finishing by pressing return.
That keyboard shortcut may or may not work the first time you invoke it, and you may need to prime the pump by invoking it as an application menu > Services > Your Quick Action name. Thereafter, if there are no other application or keyboard shortcut conflicts, it should then function as a proper keyboard shortcut.