IIRC St. Louis nearly blew the division in 2006 in the final week, each game having a chance to clinch. And they won it all.
Limping in or flying in, it doesn't matter. If anything, maybe the Angels come in have their guard down knowing we've lost 10 in a row, or have a poor record going into the playoffs. If not that, maybe they end up trying too hard not to be the streak-snapper, and start playing erratically.