Do them in pure HTML and style with CSS. That is the whole point of it.
they should look good on mobile and desktop. Checked.
they should be easy to print. Checked.
they should be compatible with screen readers for the visually impaired. Checked.
it should be quick to edit/design/publish. Checked. Sort of*.
there should be some design elements, like fonts and cell color, header title. Checked.
HTML has many, and I mean, Many features that need to be addressed to maximize them.
For example, there are some tags, like colgroups, col, captions, thead, tbody and tfoot to organize the meaning of the elements. This opens the door to optimize the user experience, for example, you can collapse unnecessary columns, keep the headers fixed, etc.
You also can easily stabilize them using CSS, and prepare stylesheets for print.
The point marked as * is relative because it depends a bit if you are using an application or pure code, if you are using a database, make them generated in an automatic way, etc. There are many ways to edit the data, but once you establish a method it is easy. That is the point at the end.
are efficient. Checked.
and accessible. Checked. It is free.
workflows... here is a specific topic that its too board to address. A workflow can be a team of persons or a simple way you work. You simply need to learn to do solid steps, and you can reuse some elements you did well in a past project.
Html world is a wide collection of things you can plug... or need someone else to plug it for you. Either you go to the store to buy a screwdriver you need, some calipers, etc, so you can make your own toolset, or you can look for a premade thing assembled by someone else.
When I say to them in pure HTML is a thing you can do. A do it yourself. And it is not a bad thing, it is not an "unprofessional" thing. I mostly do not think in premade applications.
And when I say "Make them generated in an automatic way" I think in a custom PHP script. But also this PHP script is based on the needs of the project, which in your case is not defined.
I do not think your project needs any database at all. Because you are thinking of "manual" solutions as InDesign and PDF. A custom made database will generate you tables from thousands of data, but you probably do not need more than a handful.
At the end. Your project needs to be defined further more than a post on a forum can.