Isn't the correct way to drill holes through the joists at half
depth and run the cables through? The joists are only 70mm deep,
so would drilling holes all over the place weaken them?

No, it is acceptable if the space is not likely to be used. Indeed, it
prevents the need to drill the joists, thus weakening them. As you are now
converting the area into a storage area, you'll have to solve the problem.

This can be done by drilling the holes in the prescribed manner (you should
only drill holes in certain defined parts of the joist of certain prescribed
sizes and quantities). You can also add battens to the joists, so the
flooring sits on those instead.

If your joists are only 75mm, then they are very weak. If the span is quite
large also, then the storage capacity of the area in terms of weight will be
pretty low, especially after the weight of flooring/battens is considered.
Don't start putting large boxes of books up there.

Christian.