No head of state. Useless, pointless, and a waste of resources.
Keep head of state and head of government combined.
A seperate Head of State can serve as a politically neutral national figurehead and perform ceremonial functions, thus leaving the Head of Government free to deal with controversies and be blamed if things go wrong without causing a constitutional crisis.
What's the point of having a politically neutral figurehead?
I'd rather abolish the ceremonial functions, or have the VP or First Spouse deal with them, or just let the President keep them.
Presidents make mistakes all the time without causing a "constitutional crisis."
It dilutes accountability. I'd rather go in the opposite direction, that would be a main reason why I would want to adopt a parliamentary system: so everyone knows exactly who the leader is, combine the President and Speaker and get rid of the Senate. Creating a separate head of state would do the opposite of the entire rationale behind going parliamentary.