No, Democrats did not just want to "count all the votes" in the 2000 election

Overall, there are 73 mentions of the phrase “count all the votes” in major newspapers between November 7th, 2000 and November 27th, 2000 . . . more than halfway through the recount process. Over the next 20 days, it occurs more than 250 times.

In other words, people didn’t get interested in counting all the votes when Al Gore offered a statewide manual recount as an alternative to lawsuits. They got interested in counting all the votes when the partial recounts suffered legal setbacks. People are retroactively remembering something that emerged several weeks into the recounts as having been more central to the Democratic case than it actually was. “Count all the Votes” became the central argument only after the Supreme Court had squelched the preferred “count some of the votes”.

People remember having been outraged when the Supreme Court declined to “count all the votes”. But partisan fervor was already running very, very high by late November. Odds are that anyone who is old enough to remember that election was very probably just as angry and passionate before “count all the votes” emerged as the unifying rallying cry for Democrats. Indeed, most of those very people were probably in favor of counting some of the votes before they were against it.