It doesn't adequately handle the SaaSification case, which is when someone hosts the software behind a paywall. It only handles the "Tivoization" case when someone ships the software inside a closed source box.

Are you sure? AGPL was written expressly to deal with SaaSification. Wikipedia seems to agree with me, at least[0]:

"Both versions of the Affero GPL were designed to close a perceived application service provider (ASP) loophole in the ordinary GPL, where, by using but not distributing the software, the copyleft provisions are not triggered. Each version differs from the version of the GNU GPL on which it is based in having an added provision addressing use of software over a computer network. This provision requires that the full source code be made available to any network user of the AGPL-licensed work, typically a web application."

The non-Affero GPLv3 handles Tivoization, FWIW.

[0]: https://en.wikipedia.org/wiki/Affero_General_Public_License

GPLv3 also added patent grant which is super important if you deal with software built by corporations.

Isn't the purpose of GPLv3 to handle Tivoization and AGPL was made to handle server-side sort of use-cases?

If I use AGPL-licensed software in a SaaS type of environment, I still need to release the source-code even if I'm not distributing the software directly, correct?

Correct.