"Typing in a domain name and clicking "search" on instantdomainsearch.com immediately takes you to the GoDaddy"
That is not exactly correct. Doing a search returns data from instantdomainsearch.com itself. Only once it is returned do you have the option to go to a registrar like godaddy, among others (or just take that name to a registrar of your liking), to actually register the domain.
I have used instantdomainsearch.com many, many times myself to great success.