The goal of the tiny/mini programs are very different per gym. Some do not ever have their tiny teams compete (just exhibition) as it is more about getting out on the mat in front of people and remembering what to do. Others have strong programs where you know going in, this is meant to be a competitive team from day 1, no matter age. It is the way of the gym. Most of these teams are probably in the 5 year old range rather than 3 year olds.....
For example, The Stingray Allstars (
UCA 2013 - Stingray Allstars Team Grape - YouTube ) is a gym where going in, you know that the team "Stingrays - Grape" is going to be competing. It is always a strong tiny team.
Central Jersey Allstars is also a well known gym, but their tiny team was exhibition because you can tell just by looking at the kids, it was a much younger beginner group (
CJA Tiny Exhibition Jan 12 2014 - YouTube ). 3-4 year olds have to be cat hearded more so, as some have called it & its about just getting out there and doing the routine.
Neither is "better" for all children per say, just different types of teams. Depending on the child & their age, they may crave a very serious team that is competitive, or they may shy away from the competitive aspect and just want something to have fun. All of these children though will succeed in cheer if they stick with it, no matter what team they were on at age 3-5 though.
At such a young age, some of the kids will stick with cheerleading and some may get burnt out. It just depends on the kid and their surrounding enviornment. Is mom putting them in the sport or are they the one asking to go to class everyday? Are they looking up to an older family member that also cheers? How do they handle being in front of others & performing? How do they handle their stress/fears? All factors