No you don't need to. People cannot change the license of code they don't own nilly-willy, only the owner(s) of code can (which can get hairy when a codebase has multiple owners).
However a license can allow sublicensing, aka sharing the code or software under a different license, but this right must be explicitly granted by the license (or by the code owner directly), and they may be conditions attached (such as attribution)
MIT:
"[...] Permission is hereby granted,[...] to use, copy, modify, merge, publish, distribute, sublicense, [...] subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software."
GPL:
"[...] You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License."
So the MIT license grants you the right to redistribute the code under GPL (under attribution condition), but GPL does not grant you the right to redistribute the code under MIT.
(Note that all of this applies to redistributing the code or the software, not about your own internal use of the code or the software)
There do exist people who consider the usage of open source software for cloud services without providing the derived source code also to be commercial abuse. To prevent this usage scenario is what the AGPL license was made for.
The OP is trying to prohibit commercial use, which isn’t a feature nor even a goal of the GPL.
The thing is, AGPL allows commercial use just fine, it only disallows proprietary use.
The fact that companies are dissuaded from making commercial use by being prevented from putting a proprietary moat around the software is really an indictment of those companies.
Heck, you can even make AGPL software a bit more friendly for commercial enhancements if you want by granting an exception for plugins the way some GPL software does, but those same companies would probably still be dissuaded.
Doesn't the first clause of the MIT license prevent this?