The strongest argument against signing Andrus or Iglesias at this point has to do with roster flexibility. And it's got more to do with the 26-man roster than the 40-man.
Signing one of those guys would mean we’d not only have to boot another guy from the 40-man (likeliest candidates being Ort, Brasier, Kelly, Dalbec, Duran), but it would further squeeze the roster by adding another guy to the 26 that you can’t send down.
Here’s another way of looking at it.
26-man locks (no options):
Sale, Kluber, Pivetta, Paxton, Jansen, Martin, Rodriguez, Brasier, Bleier (9)
Kiké, Devers, Turner, Duvall, Mondesi, McGuire, Refsnyder, Arroyo (8)
Yoshida, Verdugo, Whitlock, Houck, Casas, Schreiber (6)
Optionables with MLB experience:
Bello, Dalbec, Duran, Crawford, Wong, Winckowski, Mills, Ort, Kelly
The first 23 players are locks for the active roster, if healthy. You need a second catcher, so either Wong starts in Boston or you use Alfaro in Story's spot on the 40-man — either way that's 24. You need a 13th position player, and as it stands that's likely Dalbec, Duran or Andrus/Iglesias (25). Bello is arguably our best pitcher next year, so I'd personally have him up too. If everyone else is healthy, that's 26. When Story returns, that'd be 27. (I'm aware Mondesi might not be ready for opening day, but he's reportedly only expected to miss a little time if at all).
Sign Andrus, and here's what the active roster could look like in July:
Sale, Kluber, Pivetta, Paxton, Jansen, Martin, Rodriguez, Brasier, Bleier, Kiké, Devers, Turner, Duvall, Mondesi, McGuire, Refsnyder, Arroyo, Yoshida, Verdugo, Whitlock, Houck, Casas, Schreiber, Bello, Story, plus Wong/Alfaro and Andrus/Iglesias (that's 27 players)
With Andrus or Iglesias on the 26-man roster, we can't use a roster spot for call-ups without DFA-ing or IL-ing someone. Wouldn't you need at least one or two to use as a carousel for other players on the 40-man? I'd think you'd need more than one, because of the new rule restricting guys getting sent down to five times per season. Using only one spot to churn through relievers like Mills, Ort, Crawford and Kelly uses everyone's options more quickly. Signing Andrus or Iglesias would also consign Dalbec, Duran and Valdez to Worcester for the season, at least unless/until someone else gets hurt, because you can't send any of the other regulars down.
In other words, the Sox need another middle infielder. But it seems far better to use someone already on the roster — ideally the expected 26-man roster — in a trade for that player (namely: Bleier, Brasier, Casas, Dalbec, Houck, Paxton, Pivetta, Refsnyder, Sale, Schreiber, Verdugo or Whitlock — and several of those are non-starters). Otherwise we'd seem likely to lose someone on the 26 to fit that middle infielder anyway, because there are precious few players who can be sent down.
Injuries usually settle these things in the long-term, but you can't count on an injury to fit everyone onto the roster. And if there's ever a day when everyone's healthy, then someone would have to get the boot.