There might be a bug with adding modules to factories.

When I try to add a module to any kind of assemblers or beacons in factory planner, I see only speed and effectiveness modules.
When I set Productivity module as a default in preferences for assemblers and try to add new factory entry, I get a message "Your preferred module is not compatible with this machine".
When I set Productivity module as a default in preferences for beacons (which I can), it just does not appear when I create a new beacon.
When I try to add a module to any other production building (i.e. Chemical Plant, Electric Furnaces and so on) I see all three types of modules.

I can physically add modules to assemblers though, so they do work in SE.

Here are some screenshots:
https://drive.google.com/file/d/1LBPnitvTcr511lVot4knMed5koNWuU0f/view?usp=sharing
https://drive.google.com/file/d/1OG4J3n1HjWIy8gUmIaXZ1HhElx967B0y/view?usp=sharing
https://drive.google.com/file/d/1fZmytTty8QWE0jbMcDAgpWxrigXP98Y5/view?usp=sharing
https://drive.google.com/file/d/1kI4AKUugY9PQhpnpKUJ6-_9z7uR7-y5i/view?usp=sharing

I can provide any information that might help to pinpoint the problem if necessary.

Some more info: it might be related not only to a machine, but to a recipe.
For example, I can add a productivity beacon to Electronic Circuit assembler, but can't for a Manipulator assembler.
All in the same factory.

Issue with beacons persists on any of the recipes.

Just confirmed that it's not an SE problem.
Left only Factory Planner on and got the same deal:

https://drive.google.com/file/d/1AR62x8EcOBLRWT5MBk1a_g88xsf1-_41/view?usp=sharing, https://drive.google.com/file/d/1gnsB770FYlEdECZpQrz9cs1JEXjK32FM/view?usp=sharing

Appreciate the details, but you are aware that productivity modules can't be used on every recipe? This is the case in almost every mod I think, as well as vanilla of course. And they can't normally be added to beacons either, all for balance reasons. I checked out a few examples and everything seems to work the same as in the actual machine. Try adding a productivity module to an assembler producing inserters, you'll see that it won't let you do that.