Firstly, it is NEVER the machine that would cause this (if you think about it, all machines do the same basic thing - introduce water/soap solution to the drum and tumble clothes through it) - yes some machines will perform better than others etc, but the basics of a washing machine is the same.
It is how it used - the main reason we suffer these problems now and your mother/grandmother didn't is because of changes in washing habits.
The 2 main things that cause the mould is low-temperature washes and liquid based laundry detergents.
Washing in cool water all the time, will cause a build up of undissolved body oils/dirt/skin etc and the situation is made 10x worse if liquid/gel/capsule detergent is used - as NO liquid laundry product in the UK contains bleaching agents - whereas ALL Bio and Non Bio POWDERS do.
So if the user is using HOT washes every now and again and using POWDERS also, their machine will be a lot cleaner than a user who constantly uses cool washes (even a 40 degree wash is only 3 degrees over body temperature - and think of the oils on our skin).
The washing at 30 that is popular is good for the environment - but not at all good for machines and clothing (they usually need replacing sooner).
You're right about Fabric Softener, as this is essentially a grease based product, but if some hot washes are being run (as in 60 or 90 degree washes), then it doesn't have the chance to build up.