I can't believe I didn't think of that...will definitely be using that idea from now on! Thanks for posting!
Keep in mind that this will only work for Open or Members Only groups or users who have already been invited to a Private or Secret group. For Private and Secret groups users still have to be invited using "Invite people to join this group" so that they are added to the group permissions. And in that case they will receive the template group invitation via email. Otherwise if you send the separate email they'll get an error message when they try to join.
With that caveat, my team thinks AT's suggestion is a great one, and we will be following up any Private or Secret group invitations with an email that has the direct "Join group" link.