It seems that one major sticking point was the Marlins' unwillingness to agree to blanket no-trade protection for Pujols. But let's say he was willing to overlook that, for whatever reason. Let's say the Angels never jumped in the bidding, and Pujols' hurt feelings over his negotiations with the Cardinals were enough to lead him to take the Marlins' money.
It wouldn't have been enough to save the season. According to Fangraphs.com, Marlins first basemen posted a WAR (wins above replacement) of -0.6 in 2012. Pujols contributed 3.9. Thus, Pujols theoretically would have boosted the Marlins' record by four or five wins, which would have placed them ... right about in last place in the National League East, perhaps in a tie with the Mets. And then ... they might well have traded him. He might be a Blue Jay or a Dodger right now.
Perhaps the more interesting what-if is the alternate. Imagine that the Marlins decided they wanted Pujols so badly that they were the ones who conceded, granting him no-trade protection.
One possible scenario is that Pujols would be one of three name players left standing in Miami, along with Ricky Nolasco and Giancarlo Stanton. But it's just as easy to envision the Marlins going in an entirely different direction over the past four months. If they were locked in for another $180 million or more on Pujols, perhaps they wouldn't have traded away nearly so many veterans in the first place.
As for the Angels, they might have pursued another significant upgrade in the absence of Pujols, perhaps Aramis Ramirez or a reliever. But it's difficult to conceive of any combination of moves that would have equaled Pujols' contribution to the 2012 club -- never mind making the team another four wins better, to get into the postseason.
Gio Gonzalez to the Tigers: Of all the deals from the 2011-12 offseason, few if any had more impact than the one that sent Gonzalez from Oakland to Washington. The Nationals added a hurler who turned out to be a co-No. 1 with Stephen Strasburg. The A's immediately filled the hole in their rotation with Tommy Milone, while Derek Norris stepped in as their starting catcher in the second half.
Oakland shopped Gonzalez to a lot of teams, though, and there was a great deal of interest. One club that found the lefty quite intriguing was Detroit, which would have plugged him in alongside Justin Verlander and Max Scherzer in a threesome that would have been unmatched for pure stuff by any group of starters in the Majors. With Doug Fister also in the mix, the Tigers would have had a stronger rotation from day one.
In order to make such a deal, they would almost certainly have had to part with top pitching prospect Jacob Turner, who later went to the Marlins in a package that landed Anibal Sanchez. Turner has a higher upside than Milone, but he entered 2012 with a total of 17 1/3 innings above Double-A. Though he made it to the Majors this year, he surely wouldn't have stuck in the A's rotation from the start of the season, as Milone did.
Thus, the Oakland's remarkable season might well have had a tougher time getting off the ground without the stability Milone provided. Meanwhile, the Tigers wouldn't have needed to trade for Sanchez, and probably wouldn't have been able to anyway, without Turner to dangle in return. Perhaps they might still have done a smaller deal with the Marlins for Omar Infante, but just as likely they would have had to look elsewhere to add a second baseman at the non-waiver Trade Deadline.
Carlos Beltran to the Rays: Once Pujols came off the market, Beltran shot to the top of the list of the most desirable hitting free agents. He ultimately signed with St. Louis for two years, putting up numbers that in many ways rivaled those of Pujols. It wasn't a given he'd go to the Cardinals, though. The Indians were interested, and so were the Rays.
Few teams could have benefited from a bat like Beltran's more than Tampa Bay. The Rays finished 11th in the American League, and last in the AL East, in runs scored. When Evan Longoria missed a large chunk of the year due to injury, the Tampa Bay offense cratered. The Rays had too many holes in their lineup, and desperately could have used an all-around offensive talent like Beltran.
Beltran was worth 3.6 WAR, according to FanGraphs. Considering that he would likely have eaten most into the at-bats of Matt Joyce and Luke Scott, who combined for 2.1 WAR in more than 800 plate appearances, it's easy to see him being worth a couple of extra wins over the course of the season to a team that finished three games out of a Wild Card spot.
And in an extreme case like the Rays, a team that played so many low-scoring games, Beltran's added offense might very well have made even more of a difference than that. An extra 20 runs over the course of the season could very well have been the difference between playing in October and watching the postseason on TV.
The Cardinals, meanwhile, would have had to look elsewhere to bolster their club, and there weren't many available players who could have had the same impact as Beltran. St. Louis made the playoffs by an even narrower margin than Tampa Bay missed them, so it's probable that Beltran's decision tilted the playoff fortunes of both teams.