To begin, I absolutely would not stand by that, this is, to be honest, terrible advice. There is nothing faster about forwarding, in fact, forwarding mail will end up getting your server blacklisted by Gmail in many cases.
When you forward mail it leaves the senders servers, goes to yours, waits out any delays there, is processed, then is sent on to its destination and the process repeats itself.
I would ALWAYS recommend you configure your account within Gmail's -> Settings -> Accounts and Import UI
While I agree on principle. I think the issue is - when you POP mail into a Gmail account... those POP checks are only done at sporadic intervals. For example, it may have POP'd mail now, someone sends a message to the email address that is to be POP'd, and if Gmail doesn't check for POP for another 20 minutes, then it'll be 20 minutes before the message arrives in your Gmail. So there can be a delayed in this type of setup.
Having said that... I still agree with what you are saying. The risk is just too great of Gmail catching a forwarded spam message and blacklisting the server if you use a forwarder. Then not only does your forwarder not work, but anyone else sharing that server that is trying to send an email to Gmail will have trouble sending out that message.
So ultimately I do agree with POP'ing mail into a Gmail account over a forwarder - but I can also understand some of the concerns.
Course... my argument would be, if the boss only wants to check his Gmail account... he should be telling his contacts to write to him at the Gmail account. Claims that that doesn't look professional? Well... deal with it. Either check the mail with Roundcube, a real email accounts, POP it into their Gmail account, or just advertise their Gmail address and bypass their domain name entirely. Those are your free (or at least no extra cost) options.
Another alternative is to purchase a Google Suite for the domain name so that you can use Gmail for the domain name. But, to me, that's rather expensive. Something like $5/mo per email address. Which, I guess if the domain name just has the one email address - it might not be too bad. But if it's got 10 or 20 email accounts, that $50 to $100 extra per month. I guess this is a good way for Google to stem the herd so to speak.