I think the title explains it all but I am going deeper into my question anyway:
How can I make use of the Chrome\'s Omnibox [TAB] feature for my website?
As man
I've compared what you have against the OpenSearchDescription on my own site and I cannot see why yours is not working. The only real difference is that you are using POST to search whereas I am using GET. According to this page, IE7 does not support POST requests, so it may be that other browsers also do not support POST.
The one on my site definitely works in IE8, Chrome 3.0 and FF 2.0+. Feel free to compare them yourself and see if you can spot a difference: opensearch.XML