The only restriction is that you can’t (necessarily) provide TSAAS for free. To me that’s free as in speech but not free as in beer, and is well within my personal understanding of “free software”. As the saying goes, the right to speech doesn’t imply the right to be heard; likewise, the right to inspect and modify software doesn’t necessarily imply the right to profit from it.
I don't believe Timescale is Free software or Open Source. It's proprietary source-available software. I wish all software was Free software, but I don't have the fortitude to restrict the software I use to only Free software. Timescale is "Free" enough with this license that I'd consider using it in my personal workflow, or in a business environment.
If these terms are acceptable to you, that is great, but I find the unpalatable for core infrastructure I would be investing substantially to build on top of.
It restricts Customers from going to a Timescale-as-a-Service provider, but I don't see how it restricts the Customer from contracting to have a third-party support the Customer's Timescale installation (or to make changes to the codebase as a work-for-hire).
I agree w/ your assessment of their new license insofar as it being proprietary. Their old proprietary license made the product unacceptable for my use, but I am comfortable with the limitations to my rights with the new proprietary license. It has enough "right-to-repair" and ability to avoid vendor lock-in to make me feel comfortable that I could maintain a private fork of my use case was dependent on their software.
Edit:
My first paragraph was a quick-and-dirty. The child post is absolutely right. The license restricts Customers from providing Timescale-as-a-Service services.