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.
I'd quite happily pay the immigration fees myself to get into the states (currently in the UK also).
Also, employers cannot make you pay the immigration fees, it's illegal.
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.