That is not what I've been seeing lately on Amtraks website.
I'll see your anecdotal observation and raise you a statistic.
According to the US DOT, Amtrak's system-wide load factor in November (most recent
figures available) was just 48.7%. In fact, over the past decade, there's never been a month where the load factor has reached 70%. As I said, there
are certainly sold-out trains on certain routes at certain times of the year. But overall the numbers support the hypothesis that families that book away
from Amtrak will, on average, NOT be replaced by full-fare passengers.