If a visitor leaves the site from the entry page (either by the back button, a outbound link, manually typing a new address, or closing the browser) then GA will read that as a time on site of 0:00, because there is only one reference point. This is the case regardless of how much time is spent on that page.
when i visit an individual page under the "Content" section... for instance: mysite.com/webdesign/tips.html
i see the following stats:
PageViews
Unique Views
Time on Page
Bounce Rate
%Exit
$Index
There is not a measure for "Avg. Time on Site" so im not sure where you are looking... it maybe an area that can not measure time on site.
And, if i click the "top content" link under the "Content" section, i see a column of "Time on Page" that shows me the average time spent on page. I still dont see "Avg. Time on Site" for individual pages.
Back to the question though... your bounce rate is the % of users who land on that page, and dont go anywhere else by clicking anchor links, documents or any other form of navigation on your site.
that should have no bearing on your "time spent" on page or site... becuase if they land on your page, spend 1-2 minutes just looking then "bounce" out of there, they still spent time on page and should be registering.
Your "Avg. Time on Site" i can only see on the main "Dashboard". Im not sure where else you are seeing this.