You're right that children usually move at age 7 and 11, which translates as going into Year 3 (beginning of prep) and Year 7. Some also move at age 13 (into Year 9), as some independent secondaries start at Year 7 and some at Year 9.
Moving into Year 2 would mean going into the last year of pre-prep, so the most selective schools may already be full, or only have one or two spaces, which will therefore be highly sought after. Also this may well mean having to change schools again after just one year.
I have children at one of the most selective Cambridge schools, having entered at Year 3. I didn't tutor them or have them tutored, as I wanted them to only go there if they would comfortably get in - and I don't believe many people tutor their six year olds in Cambridge (unlike London!), but perhaps people just don't admit it.
Depending on which schools you are considering, there may well be Year 1 and Year 2 taster mornings, with children having to be registered sometimes as soon as possible (some schools use this as a tiebreaker), while other schools let you register them right up to the December of Year 2. The entrance exams are then in the January of Year 2, with offers issued almost immediately. Parents then have a few weeks to decide whether or not to accept. It's quite usual to enter for more than one prep, so parents may have more than one offer to consider.
The most selective schools take something like the top 20/25% at Year 3, with this reducing further at Year 7, due to demand - so, yes, it's usually much easier to get in at Year 3. Other preps are far less selective - and some not at all. In addition, the preps all have a different vibe, suit different types of children (academic, musical, sporty, etc) and attract different types of families, which for us was important too, as we wanted to feel at home at the school we chose - which we do.
If you let us know which schools you are considering, we may be able to help you further...