I have successfully changed this, but not with Google. It appears that a search engine must add some special code to their homepage to make this work. If you visit a site that has this code, then you can add that site to your search engine list. For me, visiting stackoverflow.com allowed me to add their site as a search engine. I'm sure google will have to make some change to enable this as well.
I don't know what this change is, but since it works for at least one site for me, it suggests there is something that needs to get done by the search engine.
There is a difference between the "start with" settings and the "Home page" settings. Home Page is set when you enable the home button.
It appears that the site has to support the "OpenSearch" protocol. In stackoverflows head section it lists the following:
I don't see anything similar in google's head tags...
<link rel="search" type="application/opensearchdescription+xml" title="Stack Overflow" href="/opensearch.xml">