If you take a more general view of a town it should look more or less like this:

https://steamuserimages-a.akamaihd.net/ugc/789752398597427003/AFE1661BE6451F397ADD68407A804C5635503155/

except that now unsorted mail is showing slightly red-ish. So only about 1/3th of the town buildings will be able to produce mail or waste. For them to effectively produce, they have to receive commercial or industrial goods, be inside a caption area of a cargo bay, and have an active connection (line) with vehicles able to transport mail to the Post Sorting Center. Same story goes for waste.

Your description of the situation is complete, but something is not satisfied yet. It is difficult for me to see what is missing in your setup. Or, maybe a conflict of installed mods that also alter the working of town buildings?

This is more general of my city:
https://zupimages.net/viewer.php?id=20/11/ehro.png

Here is the list of all modules that are loaded:
https://zupimages.net/viewer.php?id=20/11/0zp2.png
https://zupimages.net/viewer.php?id=20/11/5lwh.png
https://zupimages.net/viewer.php?id=20/11/qoe0.png
https://zupimages.net/viewer.php?id=20/11/qoe0.png
https://zupimages.net/viewer.php?id=20/11/jayu.png
https://zupimages.net/viewer.php?id=20/11/9siq.png
https://zupimages.net/viewer.php?id=20/11/hqk6.png
https://zupimages.net/viewer.php?id=20/11/2uzj.png
https://zupimages.net/viewer.php?id=20/11/wdko.png
https://zupimages.net/viewer.php?id=20/11/m2vt.png
https://zupimages.net/viewer.php?id=20/11/ix8j.png

There must indeed be a module that interacts, because when I start a new game with just the module "Yeol's Senseless Industries" I have the icons of mail and waste goods in the city!

I found the module in question: https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=1934909230&searchtext=building

Thank you for your excellent work.

Oops, I joined too quickly!
There seem to be several

The solution for having mail and waste is to have "Senseless" below "Town Buildings: no end year" in the mod load order. I hope this helps. ;)

I'm not sure I understand: Should I load it as a captor or last?

You can change the load order of the mods when you load the game. Beside each mod is a number. That is the load number in the sequence of mods that are loaded when you load the game. Click on the number of Senseless, and click on the double arrow down. Senseless will be thrown at the back of the load order list. Load the game, and normally you should see the mail and waste appear in the towns.

Good the situation is the following one: I launch new cards by activating the modules one after the other: I did not find other module interfering with the operation "Senseless".
Except that when I load a saved game by deactivating the "Town buildings: no end year" module, I can't find the mail and waste! Even changing the loading order by placing "Senseless" at the end. It is therefore irreversible!
It seems to me that it must be indicated on the page of the module in order to avoid mishaps for other players.

I did not test it, but I believe that when you put Senseless after "Town buildings: no end year" in the load order, all new houses will gain mail and waste capabilities. To accelerate the process, you might delete some houses, so new housing is growing back in place. But again, I did not test this, but it would be logical knowing how the mods work.

I just started a game with Senseless Industries mod and I can see mail icon in residences but the cargo is not being sent to the truck station. Does residencies also only produce mail if the city is provided with commercial and industrial goods? I have passengers connecting but no goods.

You need indeed to first deliver cargo to the town buildings for them to produce mail or waste. After that, mail will start to snowball once a critical mass has been reached.