Holiday has many meanings
It might mean "a period of time when you are not at work or school"
With that meaning, we say "on holiday" or "on a holiday" but "during the holidays" or "in the holidays"
For example, we say
I went on a road trip on holiday
I went on a road trip on a holiday
I went on a road trip during the holidays
I went on a road trip in the holidays
However, a holiday or a public holiday is countable and also means "a day when most people do not go to work or school, especially because of a religious or national celebration".
My question is that
Is it correct to say?
Children don't go to school on a public holiday
Children don't go to school on public holidays
Children don't go to school in/during public holidays
I guess we say "Children don't go to school in/during public holidays" because we use "on" for just 1 day like "on Monday" or "on Sunday" or "on a cold day".
But a week or "a month" or "a year" makes of many days and we say "in/during the week" or "in/during the month" or "in/during the year"