Teams are more likely to score at least one run than not with a runner on 2nd and no outs or a runner on 3rd with less than 2 outs based on the actual play by play results of 11 years of MLB play 1984 to 1994.
I’m not aware of any source that breaks down scoring based on whether a bases empty and 1 out situation was reached by the first batter failing to reach base compared to a 1 out HR or caught stealing or only runner scoring by balk, wild pitch or passed ball.
Checking a single season there were 29 grand slams hit with 1 out in 1989, one was a walk-off by Kevin Bass which is excluded from the results (shortened inning) and none were in extra innings which are also excluded from the linked analysis.
(13) Only scoring play in inning: Albert Belle, Daryl Boston, Ivan Calderon, Dwight Evans, Mike Fitzgerald, Glenallen Hill, Jeffrey Leonard, Lloyd Moseby, Bill Schroeder, Mike Scioscia (Aug), Dwight Smith, Tim Wallach, Matt Williams
(11) Not first but last scoring play in inning: Todd Benzinger (June), Todd Benzinger (Aug), Rob Deer, Bill Doran, Kent Hrbek, Ron Karkovice, Steve Lyons, Louie Meadows, Eddie Murray, Mike Scioscia (June), Ernie Whitt
(3) Not first or last scoring play in inning: Carmelo Castillo, Fred McGriff, Garry Templeton
(1) First but not last scoring play in inning: Terry Steinbach
So 4 out of 28 times or 14.3% the teams scored after the Grand Slam when over a longer time period and 9,653 times larger sample size teams scored at least one run 15.9% of the time with 1 out and the bases empty. The reason 14.3% and 15.9% are similar is called a fluke, it’s way too few. But if you randomly select enough events it becomes increasingly likely that teams score close to 16% of the time with 1 out and the bases empty.
Mike “Rally Killer” Schmidt in his career has 75 plate appearances with 1 out and the bases loaded. He hits 4 HR and every time his team doesn’t score for the rest of the inning, also a fluke.