I work in daycare and no we don't apply antiseptic spray. Have a chat to your team of educators if this is an option for their centre.
Nope they don't. And they're not allowed to. What if your kid has an allergic reaction to it? If you feel strongly about it, fill in a medication form and get a chemist label.
Daycares cannot apply anything that could be classed as medication without direction and instruction. You could possibly provide your own, however it would likely need a chemist label with explicit application instructions.