Biden's remarks did not explicitly say why one image was previewed ahead of the others (or why this was the image chosen), but we can speculate.
First, NASA press events are not a mainstream outlet that many people go to look at. Anecdotally I found it rather difficult this morning to actually find the press conference -- outlets like CNN and the New York Times had their own "live coverage" of the event with their commentary alongside the images, but did not actually link to NASA's broadcast.
A briefing from the White House press room is much more direct, and will reach a larger audience. Something as momentous as Webb's first image deserves a large stage.
Secondly, it is a traditional sort of honor in the US to be the subject of a speech from the president (think of a medal ceremony), and indeed parts of the speech were used to commend the Webb team:
I want to thank the team at NASA for once again showing that that’s who we are — that’s who we are as a nation: a nation of possibilities.
Thirdly, it is an opportunity for the administration to stump for STEM education and science outreach (and who's going to object to that?):
That’s why the federal government must invest — must invest in science and technology, more than we have in the past.
These images are going to remind the world that America can do big
things. And they’ll remind the American people, especially our
children, that there’s nothing beyond our capacity — nothing beyond
And fourthly, it is an opportunity to celebrate the relationship with international mission partners. A call-out from the presidential stage sends a clear message of "the US as a whole values this collaboration", while a call-out from a NASA event would not.
So, what an incredible team — joined, by the way, with our international partners, the European Space Agency and the Canadian Space Agency. So this is an international endeavor.
All in all, I think the decision to have a ceremony is obvious, and having a ceremony without any images would have been just cruel to everyone who was excited!