That's right, but the fact is that the president's goal is to get everyone who can receive the vaccine to be vaccinated. He lacks the authority to issue a simple vaccine mandate for everyone and is just doing what he does have the authority to do. The vaccine is not just about workplace safety and people who work from home should still get the shot.
I call this "late fee" logic. Often times an apartment will want to charge $X for rent but only charges $Y. They make up the difference by imposing fees and other restrictions so by the end you are basically at $X. I hate these policies because I hate people lying to me. Cut the shit with these late fees and just raise the damn rent.
Likewise, if the goal is to vaccinate the entire country, don't try to beat around the bush and have OSHA do it, actually mandate it.
I'm not in favor of vaccine mandates, I just hate when poeple tell me to my face "Oh well we need to do this because of A, B and C" where everyone knows it's really "We're doing this because it will accomplish X, Y and Z"
The president does not have the legal authority to make such a general mandate. Congress might be able to do so, but congress works slowly and this needs to happen as soon as possible.
Like it or not, that is the system we have in the US.
"the legislature is too slow" is a purposeful design to avoid tyranny of short term thinking. When you hear the executive proclaiming this it never results in more civil liberties.