Yes: routers do many things.
The new 82U router must be configured with all of the rules necessary for OP's network.
That includes the allowed DHCP IP address range, the static IP addresses (MAC reservations), port forwarding, security, parental controls, lease times, logging, etc..
All current network devices have been configured to go to 192.168.1.2 as
the router with those configurations.
If the network devices somehow find and/or go to 192.168.1.1 they will not get the necessary information. Or even incorrect or malicious information....
And allowing client devices to make such adjustments based on some discovered router would be a serious security matter.
= = = =
Footnote: with both routers being ASUS the original router's configuration settings may able to be saved and exported to the new router.
https://www.asus.com/support/FAQ/1001376/
I think you have static IP address confused with DHCP reservation. They are not the same thing.
A static IP address is something you manually configure on the client device. You have to configure it OUTSIDE the DHCP allocation range. In this case the OP ended their range at 250 and so they assigned a manual static IP address on their client of 251 to be just outside their DHCP server range. When you set a static ip, it's done on the client side and you have to specify the IP address, gateway, dns and netmask yourself.
A DHCP RESERVATION is simply a pre-defined assignment of an IP address to a specific MAC address. Meaning, the DHCP won't give away the specified IP address to any other client other than the MAC address specified.
When a client first boots up and connects to a network, the client will send out a DHCPDISCOVER message. The server hears the discover message and will give the client a configuration(IP address, DNS, Gateway and netmask) that'll allow them to connect to the network and the internet. If there is a reserved IP for that client, the DHCP server will assign it, if there is no reservation the DHCP will assign it another IP address within it's allocation range, in this OP's case somewhere between 192.168.1.2 and 192.168.1.250.
That said, with a DHCP reservation, you're only predefining the IP address. The DHCP server will still give it the same configuration information for gateway, dns and netmask as it does to other clients. So once your reserved client boots up and connects to the network, it will send out a DHCPDiscover message. The DHCP server will respond with it's IP address, gateway, dns and netmask, the same as other clients. The only difference is it will get the same IP address every time.
In a typical home router, the IP address of the router/gateway is 192.168.x.1 and the DHCP allocation starts at 192.168.x.2 or 192.168.x.100 depending on which router you buy. The allocation range ends with whatever range you select. If you want 50 clients max, it'll be 192.168.x.2 to 192.168.x.52 etc....
In the OP's case, they said their DHCP range is 192.168.1.2 to 192.168.1.250 so their router has to be outside of that range. So you set the router to 192.168.1.1. Most routers will never allow you to assign the router IP within the DHCP allocation range.
In the OP's setup:
Router/Gateway: 192.168.1.1
DHCP RANGE: 192.168.1.2 to 192.168.1.250
Static IP possible range: 192.168.1.251 to 192.168.255