I really did not feel that. I just graduated in the UK and will move to the Bay area in fall (when H1B starts). All the companies I talked to were totally ok with H1B.
Essentially if you aren't Intel/MSFT/IBM you don't stand a chance of getting an H1B application through
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.