By modern convention, the VP acts as the Senate's chair only on ceremonial occasions or to break ties
However, before the 1950s, VPs acted as chair most of the time and in that role were active participants in debate although the exact amount differed depending on the individual VP. VPs of this time were not actually involved in the President’s administration and were considered to primarily a part of the legislature rather than the executive.
This changed following World War II, with VPs becoming a politically active part of the executive and stepping back from their role as President of the Senate. The role of the President Pro Tempore, who by convention is the senior person in the majority party, Has expanded to fill the void, although Charing the Senate on a day-to-day basis usually falls to a nominee of the President Pro Tempore, usually a junior Senator from the majority party.
Very little of how the Government operates in practice is actually in the Constitution, for example, the committee system of Congress is not in the Constitution. Apart from the existence of the roles, and a very short list of ceremonial duties for the VP, almost all of how the Senate actually organises itself is up to the Senate.
So, of course the founders had no idea how this would work 250 years later. For example, the original idea was that the VP would be the runner up in the Presidential election, i.e. the President’s biggest rival. That worked about as well as a chocolate frypan.
The system was designed to be deliberately loose so that future generations could shape it to their needs. That’s why it’s lasted so long - the median age of Constitutions around the world is 7 - France is on its Fifth Republic with 3 monarchies and 2 empires since the founding of the USA. Prescriptive Constitutions stop working and get abandoned, permissive ones last.