This is just plain wrong in my experience. Even small companies (< 15 employees) can easily get an H1B petition approved if they apply before the cap is filled and pay a lawyer ~$2k.
This hasn't been true since the recession. Also when the cap limit is hit a lottery system is used to select petitions. No companies have preference.
In fact the rules now are that they will include any petition received within the first five business days in the lottery. The rules are a little more complex than this, but it's essentially correct.
Any reasonably good lawyer knew they had to get things submitted on the first day due to cap uncertainties.
