My question is, what is the reason for this change over generations? What are the benefits of waiting until they are 2/3?
The reason is a combination of disposable nappies and working mothers.
A friend was driven nuts by her Mil who claimed hers were potty trained by six months. But this lady had to draw water from a well, boil nappies on a solid fuel oven, put them through a mangle and hang them on the Aunt Sally above the stove to dry.
Much easier to sit baby/toddler on a potty regularly. This is why, in times past, little boys wore dresses like their sisters (also all children wore same clothes passed down to save money). The dresses could easily be pulled up for access to the nappy and sitting on a potty.
Cloth nappies need changing regularly as they soon get sodden and leave damp patches wherever the child sits. I remember seeing children on a paediatric ward with shocking nappy rash covering the entire bottom, vulva and pubic area up to the waist due to sitting in a sodden nappy for long periods. Disposable do what they say, absorbing the liquid away and leaving the bottom dry. You never see nappy rash like that now (except in cases of extreme neglect). On the other hand disposable nappies are a shocking environmental disaster as they do not degrade.
By starting the child young they get used to using a potty and if you do not use disposable you are saving the environment. They still won't be able to use the potty/toilet independently until 3+ it takes a lot to realise the need to go to the loo, remove clothing, go, pull up pants etc, wash hands.
But I do think it helps start regular habits, especially bowel. And if you wait until they are 3 some children will look at a potty and say "Nah" and quickly learn it is a way of really winding their mothers up. But a child that has sat on a potty since they were tiny will already be used to going on one and it's never an issue.
A friend used cloth for all hers but they needed changing through the night as sometimes they'd leak and the bedding needed changing too, and also they got cold and soggy waking the child. I used cloth by day and disposable at night. My babies both slept through the night by three months, I wasn't going to change that. But disposable crept in during the day too - the childminder wanted them. And if we were going out etc.
Eldest was dry at 2 1/2 - we were at toddler group and went to the toilet to change nappy where she saw another child on the potty. "I could do that" she said and she did. Within a few days we had dispensed with nappies, and nights too soon after.
Youngest was harder. She just peed wherever she was. Lots of wet patches on the sofa. But it was summer. We went for a family walk one day and within five minutes she had wet herself and was walking like a cowboy. "Haven't you got anything to change her into?" said the husband. "No. She's got to feel the discomfort" was the reply. It did help and she soon got the idea.
Children felt the discomfort of a cold wet cloth nappy and that also helped. They are quite happy in their disposable nappies and don't notice feeling wet.
I'd start sitting your child on the potty at regular times like when they wake up and after meals soon, it is timing as pp have said but they get used to it and will catch on sooner. And you will be doing your bit using less disposables over time.