Correct me if I'm wrong, but the President is not even required to give press conferences in the first place. He could simply record a statement and release it.
He's also not required to give a speech when it comes to the state of the union. Article II, Section 3, Clause 1 dictates that the President must "give the Congress information on the State of the Union". Nothing says that it has to be done in person. For over 100 years from the time of Jefferson to Wilson, messages were sent to Congress.
I see no Constitutional requirement whatsoever about the President being required to do any type of press conference. In fact, while he can do what you said, there's nothing that requires even that for the press.