Landing at 9:15pm after a long flight from the US with a kid, I’d absolutely do an airport hotel for the first night.
By the time you clear immigration, get bags, and get into the city, it’ll be late. Instead of rushing, dealing with check in, and paying $1,000 for a hotel you’ll barely use, sleeping at the airport lets everyone shower, crash, and reset.
Changi airport hotels are comfortable and very family friendly. You’ll wake up rested, have breakfast, and then head into the city fresh the next morning. It usually saves both money and stress in exactly this situation.
If cost is a big factor, the airport hotel for night one and a nicer city hotel after is a really smart split. You’re not missing out on anything late at night anyway.