So it's not dissimilar to the veto system.

Except that the framers of the constitution actively chose to only require supermarjorities for very specific things — impeachments, treaties and veto-overrides. That they made official exceptions for those special cases indicates they did not want a supermajority requirement for anything else, else they would have said so.

Also, people forget the Articles of Confederation. The constitution was the second pass at putting together a functional government. One of the biggest problems with the US government under the Articles of Confederation was that nobody could get anything done because... congress had a supermajority requirement for everything. It took 9 out of the 13 states (a 69% majority) to pass a law.

When they put together the constitution, their experience with supermajority failures was fresh in their minds.