Following a lead at @pygo's comment, I found that my mistake was to try to read the data as
df['Date'] = pd.to_datetime(df['Date']).dt.date
This would be, as this answer explains:
This is because pandas falls back to dateutil.parser.parse for parsing the strings when it has a non-default format or when no format string is supplied (this is much more flexible, but also slower).
As you have shown above, you can improve the performance by supplying a format string to to_datetime. Or another option is to use infer_datetime_format=True
When using any of the date parsers from the answers above, we go into the for loop. Also, when specifying the format we want (instead of the format we have) in the pd.to_datetime, we also go into the for loop.
Hence, instead of doing
df['Date'] = pd.to_datetime(df['Date'],format='%Y-%m-%d')
or
df['Date'] = pd.to_datetime(df['Date']).dt.date
we should do
df['Date'] = pd.to_datetime(df['Date'],format='%m/%d/%Y').dt.date
By supplying the current format of the data, it is read really fast into datetime format. Then, using .dt.date, it is fast to change it to the new format without the parser.
Thank you to everyone who helped!