If we're looking at a going-forward basis as the primary motivator of the different attitude, companies like Google would be more allergic to the regular GPLv2 than they are. The GPLv2 license automatically terminates for a given licensee upon violation, requiring the rights holder to take action to prospectively restore their license if they want to be able to legally distribute the software in the future.
Nobody avoids GPLv2 software due to this clause. Affero's original version of the AGPL (v1) based on the GPLv2 has the same clause, nothing harsher. The currently common version of the AGPL (v3) published directly by GNU/FSF shares the GPLv3's much less harsh termination provision, and indeed companies like Google do sometimes allow use of GPLv3 code. (Those companies like Apple which are opposed to GPLv3 have issues with different provisions, not this one.)
The difference between the MIT license and the AGPL, whether it's unreasonable FUD or reasonable caution by lawyers, is not about the difference in how violators are treated going forward.
By the way, in my reading as a non-lawyer who nevertheless previously attended part of law school including the contract law course, the AGPL doesn't give any specific right to _demand_ the source code of any party any more than the GPL does. It just forces the parties who act outside the license to accept treatment as copyright infringers, as with any other unilateral license, including the damages and injunctions (and sometimes criminal convictions) that can lead to. Companies can pick their poison.
Maaaybe some jurisdictions would analyze this differently as a contract that the company agreed to, with the option to order specific performance of releasing source code. I'm not 100% sure. Again, I'm not a lawyer. I don't view this as likely if the company doesn't somehow indicate to the licensor / court / public that it agrees to the license, beyond the mere fact of acting in a way that would otherwise infringe copyright.