Isn't there some overlap between NAT, DHCP and DNS, they feel like they should be the same thing?
Absolutely no overlap. Now, can a conventional home router do all those? Typically NAT and DHCP, but not too often DNS (except to point at a real DNS). That's like saying "isn't there overlap between milk and eggs? I mean, sure, they deal with animals and stored in the fridge, but used for different purposes."
Because DHCP is assigning IP addresses, DNS is looking up IP addresses and NAT is saying that IP address points there 👉🏾
Sure, they deal with IP addresses and some devices (home routers) can do 2-3 of those fuctions, but that doesn't mean they overlap. I can read a book (IP) and my book (IP) can be stored in a book bag or on a book shelf, but that doesn't mean myself, the bookbag, and the bookshelf overlap.