Since I wasn't able to find the solution to my problem by reading this thread, I figured I should post this short reply in case anyone comes across the same issue with the stopwatch not autoloading the time.
I managed to apparently solve the "AUTOLOAD LAST TIME STAMP WHEN OBS STARTS" and will share what I did.
On OBS, I clicked on 'Edit Script' on the Scripts window, and opened it with Notepad.
On the top bar, clicked on 'Edit' then 'Find', and searched for the following statement:
if load_saved_time then
You should see that the next line of code starts with two hyphens (--), which I think that is the equivalent of writing a comment in Lua, like the following:
--update_prop_settings_current_seconds( current_seconds ) -- update current time to last time in properties
Here's what I did, though: I simply removed the two hyphens (--) from the start of that line of code, so that it wasn't interpreted as a line of comment and then ignored, so it should read as follows:
update_prop_settings_current_seconds( current_seconds ) -- update current time to last time in properties
Then I saved, closed both Notepad and OBS windows and voila, next time I opened OBS the problem was fixed for me.