Well, not to get into this debate again, but I've had purchasing agents at large companies tell me specifically the do not respond to free email addresses.
Some people swear by them and say there's nothing wrong, but when a corporation's buyers tell me they don't deal with free email accounts, I don't give them free email addresses. Their perception is that if you can't be bothered to have your own domain name that handles email, are you really a vendor they want do put their trust in with their products or needs.
Many fly by night and part time people that work normal full time jobs in some other business love to use free email addresses. So using your own just helps show that you aren't in the same category as those suppliers.
I also agree, I'm not sure there's much value at all on putting an email address on a sign. Web address for sure, but not so much email. I can't think of a single time I've ever pulled an email address off a billboard, but I have pulled web addresses many times.
All just my opinion, you might disagree 100%.