I have the same thoughts for my son. He just turned 4, but I am contemplating what you are... for when he turns 5.
Anyway here is my personal observations and info I got from Teachers at my daughter's school:
1) In my daughter's Kinder class, about 3/4's of the class were kids who were 6 or turning 6. My daughter was almost 5. She was one of the youngest and born late. But she did fine.
2) MANY parents, as an alternative to holding back their child, enters their child into Kinder at turning 5 or 5. THEN, they 'request' that their child repeat Kinder. LOTS of parents also do this. At Kinder level.. it is not such a big deal. The school does honor the parental request... and with the Teacher. It is done for emotional/maturity reasons and the child's cues. It is not a stigma.
According to a couple of Teachers I talked to personally, it is common.
3) You can enter your son into Kinder at turning 6 or 6 years old. This is also done. Meanwhile, for now... at home... home-school him. Enroll him in fun extracurricular activities... .anything that HE enjoys. To get him socialized.
4) There is a BIG difference between a "shy" child and a "timid" child. So discern that difference. Being "shy" is NOT a bad thing.. my daughter is shy... BUT, she is a VERY wise and mature and self-confident kid. And has no academic nor behavioral problems.
Ultimately.. also check with your son's school that he will be attending, to make SURE that a child CAN enter Kinder at 6 or turning 6 years old. Some schools don't. Some do. See what the 'rules' are for your son's school.
IF some kids are 6.... whether or not they had Kindergarten... SOME schools will automatically enter the child for 1st grade. Because, Kinder, in some states, is NOT a 'prerequisite' for 1st Grade.
So, check on that.
BUT.. if a parent requests... that their child enter Kindergarten at 6 or turning 6.... you can request that. But check with your school first.
That is what I did. That is my 'back-up' plan.... just in case, I decide to enter my son into Kinder at turning 6 years old. He turns 6 right at the cut off date for my area.
If I feel fine about it and per observing my son's 'readiness' for Kinder throughout this year until he turns 5 years old... then I will enter him into Kinder, at 5 years old.
We cannot afford Preschool for him... right now. Thus, that is also my concern... re entering him into Kindergarten and at what age.
My son is fine developmentally and socially. And I am home-schooling him now. But, I just have a gut instinct thing that sending him at/when he is turning 6 years old, would be better for him. My son is a tall kid.... but that does not matter to me. LOTS of kids in my daughter's Kinder class was TALL for their age. It did not impact the child.
My son, although very bright... just couldn't give a rip... about 'school' and learning.... and will not even 'sit' at a desk when we play school. Versus, my daughter, at only 3 years old, was doing workbooks already AND was able to sit at a desk and do 'school' (I home schooled her), for more than 1 hour.
Just some thoughts, as I have wondered the same thing,
All the best,
Susan