Koepel:
The I2C bus is not meant for that. The I2C bus is not that kind of "bus". This is something for Serial or RS-485.
Well, I and other have managed to use I2C over several meters. I did one test with over 20 meter cable (An uno controlling a 1604 LCD at the other end). Worked fine for days.
Koepel:
I think TCA9546 amplifies the signals. That means that the main I2C bus and every sub-I2C bus needs to have pullup resistors. You also have to check all the pullup resistors to be sure that it is not too much pullup.
The RTC has 3k3 pull-ups (controller side of the TCA) and each of the test boards has 4k7 pull-ups. and the ADC board has 5k6 pull-ups (both on each channel of the TCA). So pull-ups should be fine.
Koepel:
I see a lot of flat-ribbon cables. Are there cables with SDA and SCL next to each other ? You should not do that, not even for 10cm.
When the flat ribbon cable is used for other signals, that might be a problem as well. With such a project, the flat-ribbon cable should have GND wire between the signals.
SDA and SCL is not side by side. I have used longer flat ribbon cables before without any problems. The longest I have is about 50 cm.
Koepel:
See page 60 of the I2C standard: https://www.nxp.com/docs/en/user-guide/UM10204.pdf. The SDA and SCL should not be next to each other.
The Sparkfun Qwiic bus: they did it wrong.
The DFRobot Gravity I2C bus: they did it wrong.
The Adafruit Stemma I2C bus: they did it wrong.
You see the problem ? They all got very excited by the word "bus" and they didn't think about the weak and wonky signal that I2C is.
I know about that, yes 
Koepel:
Is everything on the I2C bus with 5V ?
Yes.
Koepel:
Can you try that empty sketch (with only Wire.begin) and measure the shortcut current for every part of the I2C bus for both SDA and SCL. One before the mux and four after the mux, that is a total of 10 currents.
When they are <1mA, then the bus is too high impedance.
When they are >3mA, then it is not according to the I2C standard.
In the end, your project might not work. The I2C bus is not meant for that.
That was a useful tip. I will do that.
Everything works fine until the heating power supply is connected.
I just replaced the PWM for the heating supply with a laboratory-style power supply (not a PWM based). It seems to be more stable. Only one "RTC time error".
So I think the problem is noise from the PWM.
The final project will use ring core transformers to power the heating wires.