Since this works 100% reliable with the NRF24L01 off of the PCB using jumpers, I know it is not related to bypass capacitors.
With the NRF24L01 off the PCB the current flowing through the ground/VCC traces is less and switching noise is less, so less filtering is needed, without any bypass caps as soon as you add the module the ground/power quality gets worse and that maybe crashing or glitching things.
Currently you have no bypass caps. And adding them is one of the primary rules, "Thou must add decoupling caps on VCC to GND near power pins for any IC/Modules"
Add a 1uF or similar between the VCC/GND pins for both NRF module and ADS module. should be able to solder it between the two through hole pins.
Another issue I noticed is that you have the temp sensor powered by VIN on the ESP, pretty sure that VIN is a 5V input pin to power the ESP, not output.
If your temp sensor is working its probably getting powered from leakage current through diodes, not from an actual power rail.
(If VIN is also USB5V output then the temp sensor will only work correctly when USB is connected, but i think it's an input only.)
You don't seem to have any I2C pullups on your PCB, do any of your devices/modules that use I2C have their own i2c pullups? They are required for I2C to work correctly.
If it still doesn't work, try adding a 100ms startup delay in the ESP code, perhaps the NRF or ADS module is on the verge of not being ready for communication at boot and the ESP is trying to communicate with it a bit too soon and causing intermittent glitches at startup.