The 22nd wasn't seen as an issue until Roosevelt was in office for four terms. The framers NEVER saw elected government office a a career path, it was usually one or two terms and out. The initial politicians were paid for their expenses, they met occaionally and returned to their homes, to run farms and other businesses. If they were to come back and see the quagmire the US government has become, they would pass out.
Also, at that time and for many elections after, POTUS and VP didn't run as a team, POTUS received the most votes, VP came in second.
All they need to do to make something happen, or not- use the word 'shall' or 'shall not'. That word carries a lot of weight and it was used in the original language- the 2nd Amendment is an easy example where it contains "shall not be infringed". (I only used that because it's easy to remember due to all of the debate, not to start something)
I have heard "The law requires specificity" many times- if language isn't included, lawyers WILL argue for and against it, probably endlessly.