There has always a difficult compromise to be found. One way to solve the problem is to use a metal enclosure and by the help of filters to shunt all the cables' EMI to the enclosure's ground, which plays the role of radio frequency plane then, if your load cell amplifier gets its signal from the outside world.

Of course, it makes no sense to put very sensitive inputs in close distance to the mains input. But nevertheless you can minimize this distance, without compromising the performance. You can for instance route all the cables in such a way to the board, that the cable entries are all at the same edge of board in rather close distance to each other. But on board you then route the sensitive signals to individual sections (amplifiers), which can be far away from the transformer and its hum. At the cable entries, there must be filters of course, which prevent the EMI from running across the board.

The idea of this single cable entry "point" is,...

1. ...that the cables leave the board from the same place, so that only negligible common mode noise can develop between the individual cables and radiated EMI is minimal.

2. ...that, by design, no EMI can run across the board, just because there exists no path between any two cables along which EMI would run across the board.

Whatever you do, you must prevent EMI from entering and running across your board. How you achieve this is your decision.

Kai