The British Parliament had already been around for centuries by their time. I don't recall hearing them (in the history I do know) complain about Parliament itself, just that they had no representation in it. Did they find a parliamentary system lacking in some way?
Followup question: why was the title "President" kept when the Constitution established a true head executive, rather than an actual presiding officer (a secretarial, rather than directorial, function), as was more accurate during the days of the Continental Congress?
The US President's role is far more powerful than merely "presiding" over a deliberative body. By comparison, the US Vice-President is accurately described as "President of the Senate", but the US President doesn't "preside" over any body the way the VP does, they actually get to make (some types of) unilateral policy, and are the boss, rather than first-among-equals, of their cabinet.
If appeal to republicanism was desired, why didn't they choose any one of historical republican-like terms, such as Chief Executive, Consul, or Director? Or, alternatively, Secretary-General (since the other cabinet members are called Secretaries)?