This is going to be a tough one to answer, however, I'll take a stab at it.

Lets start out with the timeline here, the big events are the landing and the first walk. The walk was filmed, there was no live video of the landing

You can see the landing happened at 4:18 PM EDT and the first walk happened at 10:56 PM EDT, so you would think 'Hey, perfect time, people are getting off work to the news it landed and going to bed to the sight of Armstrong on the moon', but, look at one of the main events between the two 'Decision made to proceed with EVA prior to first rest period.' The flight plan had originally called for a 5 hour rest period that would have put the first walk around 4 am, which not many Americans might have been up.

So why did they forgo the rest? According to Aldrin here,

"We had discussed, among ourselves, the possibility of evaluating, during this first 2 hours, whether we wanted to go on with the rest period (scheduled to begin at 104:50) or to proceed with the EVA preparation. I think we had concluded before the end of the simulated countdown that we would like to go ahead with the EVA and it was sometime in here that Neil called to ground and let them know that."

and Armstrong from the same source

There were two factors that we thought might influence that decision. One was the spacecraft systems and any abnormalities that we might have that we'd want to work on; and the second was our adaptation to one-sixth g and whether we thought more time in one-sixth g before starting the EVA would be advantageous or disadvantageous at that point. Basically, my personal feeling was that the adaptation to one-sixth g was very rapid and was very pleasant, easy to work in, and I thought at the time that we were ready to go right ahead into the surface work and recommended that.

So it appears, the timing just seemed to be lucky coincidence.

Having worked at NASA in an internship and been really into the space program for my entire life, I will add this, at NASA, safety is 100% paramount. There were plenty of opportunities for things to go wrong, for example, on Apollo 14, the Lunar Module encountered an error and essential had to orbit the moon several times before landing, delaying everything. If NASA or the crew decided they needed that rest after all, they would have taken it even if meant delaying the walk until 4 am.