No a child cannot go to kindergarten if they are not potty trained.
Unless the child has a special needs problem or is urinary incontinent and cannot use the toilet or has trouble with bladder control and needs to wear diapers then they must be potty trained before going to kindergarten.
If the child is urinary incontinent or has a disability and needs diapers then they can go to kindergarten wearing diapers and the nurse will change the diaper.
But if they are just not potty trained then they cannot go to kindergarten wearing diapers.
Although some kids are potty trained by 3 years old there's nothing wrong with a 3 year old not being potty trained and still wearing diapers.
Some kids may not be ready to potty train until 4 years old and I would not be concerned with the potty training unless the child is above 4 years old and getting near 5 years old and still not interested in the potty.
No 3 years old is not too old for wearing diapers whether full time or just in the night time.
Actually there really is no age that is too old to wear diapers as long as the child or person needs to wear diapers then they should wear diapers when needed.
Diapers are made for all ages and as long as you potty train the child they can wear diapers as long as they need too.
Some 3 year old children are potty trained and out of diapers in the day and some children are not ready to potty train until age 4 years old.
The child should eventually begin potty training on their own and decide not to wear diapers anymore.
However some children may not do that and may have to be forced to potty train if the child is 4 and not wanting to potty train.
Some kids are potty trained as early as 18 months old and some do not potty train until 3 or 4 years old.