I feel a little sheepish for not considering this particular ramification to the Ortiz signing - with a full roster, the Twins had to get rid of somebody when they signed Ortiz to a Major League contract. I was behind the curve on that one . . .
As for whether this is a significant lose for the Twins - it really depends on who you ask. Seth liked Romero's upside and considered him a significant prospect. I'm a little cooler on Romero. He's young (just 23), so there would have been time for him to improve. But in 71 games at AAA last season, he hit just .250 and demonstrated no power. His .300 OBP wouldn't exactly have won him any awards either. One year, especially a player's first year at AAA, does not a career make, and it's certainly possible that Romero will blossom into a Major League caliber ballplayer. But, while I'm disappointed in a general sense to lose a prospect, I wouldn't call myself heartbroken.
As for who else could have been removed from the roster, there isn't a lot of fat to trim. Luis Rodriguez was just off - and then back on - in the blink of an eye, and the Twins seem to expect him to contribute at the Major League level again this year. Taking him off of the roster would have just created a need for another move after Spring Training. Chris Heintz could have been removed, but if he were claimed it would have left the Twins even weaker than they already are in organizational catching depth, so that wasn't really a great option either. As much as I don't want to see it happen, I almost would have suggested removing J.D. Durbin from the roster now and being done with it - there is only about a 5% chance (at best, and being generous) that he makes the Major League team, and since he's out of options he'll be going on waivers anyway at the end of Spring Training. Since we're likely to lose him then, why not bite the bullet and make the move now, thus keeping Romero within the organization?
Ultimately, the guys in the front office know more about these prospects than I do. I assume that they looked at all of their options, and decided either that Romero wouldn't be claimed, or that they wouldn't be too heartbroken if he was. It will be interesting to see how he does for Arizona this year.