Welcome to the Community. I have a few follow-up questions here:
What kind of artists/tracks are you searching for, can you please provide some examples?
If you perform a search for the same artist immediately again afterwards, do you still experience a slow search?
Since you are using an ASUS router, can you confirm that you have set “Enable Multicast Routing” to On in your router settings? This may not be related, but worth to check.
Does the same behavior occur when you perform a search from the Core itself? Or does this behavior just impact your Roon Remotes?
Is there any change in behavior if you reboot your Roon Core?
What is your internet download/upload speed? You can check this by navigating to speedtest.net.