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No, this is incorrect. Just because the original code is GPL, doesn't automatically make it legal for any modifications to be released to the public as GPL without the copyright owner's consent. (In this case, the company owns the copyright.)

IANAL, but I don't think that's how it works. The GPL allows the licensee to redistribute the code and so on. But if you're working for a company, you are not the licensee, the company is (in the same way the company owns the copyright on your work, not you).

Otherwise, the AGPL would never have seen the light of day.

Modifications made to non-distributed GPL code are not automatically GPL. Only distribution of code derived from GPL-licensed code requires a GPL license for that derivative. But even on publication a modification/derivation of GPL-licensed code is not automatically GPL. It can also just be copyrighted code published in violation of the GPL.

So, it's also not true that "any employee of the company can publish the modications to the GPL-licensed code".