He wants to enforce a @dnalounge.com for all internal emails rule, but have the underlying tech be more flexible.

Actually, makes sense, because he can then save those emails on the server, back them up, and they'll be in his control if an employee ever leaves. It's good corporate policy. I believe the White House is having a similar conversation lately.

He does create a work email account for his employees. He doesn't want to force them to use a separate email client so he allows his employees to set up forwarding from his server to their personal gmail account. They can use his smtp server from gmail for sending stuff out so they get to use both their personal and work email from the same interface.

I've never used an email client that didn't support multiple accounts.

So again, I'm not sure what problem he's trying to solve here. To be clear I'm not saying he shouldn't be able to set up his email this way and have it work, just that I don't get why he WANTS to do it this way. What is it accomplishing?

Gmail.com is this email client. You can't add an external account, you have to forward.

gmail.com supports fetching email from external servers via POP (it's under settings, accounts). His users could use that and skip most of gmail's spam filtering completely. It doesn't (reliably) support forwarding mail to it without using some list software in the middle. I believe if jwz set up mailman or something similar, it wouldn't have a problem with that, because mailman would properly re-write the envelope sender as the list address.

You can add external POP accounts, and I think that would be the solution for OPs problem.

Neither am I.

Especially since you can configure gmail to serve as an interface to arbitrary third party POP3/IMAP services.

Which he mentions as being a viable but undesirable option.

"This would work but I'd rather find a more elegant solution" seems like a perfectly reasonable desire to me.