I agree with previous posters - good schools do take into account the age differences at every stage and will normally do assessments separately for each age group. One headmistress told me that the differences induced by age usually even out by the end of pre-prep.
As for the assessments, I also agree that the idea is tutoring a three-year-old is odd and actually quite embarrassing. I certainly didn't get my DD tutored, but I get the impression, both from people I've met and previous threads, that a lot of families do it. If you decide to get your DD tutored, I would recommend someone who is familiar with the schools you are applying to and knows what they expect (and may also act as a 'recommendation' to the school), otherwise it's not worth it.
In my small experience, the assessments range from normal/nice (checking that your DD is sociable and enthusiastic and has basic literacy and numeracy skills along with a good attention span) to plain weird (looking for fluent readers/writers, and I am not joking).
It's a bit of a cliche, but do make sure that the ones you go for are right for your DD. I think that if you start tutoring at age 3, your DD may well end up in a very competitive and high-pressure school, which may or may not suit her (and you).
And as Allag said, places do come up all the time and I am still to meet someone who didn't get a place when willing to pay. But of course the competition for the best schools is just ruthless.