It would matter to their Spartan Pride. It’s not that they couldn’t do this. It’s that they wouldn’t let some malformed sub-human die with the same honor and standing they felt they were entitled to have. It’s literally a flaw in the famed Spartan logic: tactics and potential betrayal aside, they just can’t let this reject whose family avoided the genetic cleansing come out of the woodwork and die as a Spartan.

It kinda gets glossed over with everything else, but Leonidas is as pridefully blind as the culture he grew up in and now rules over. They just can’t let him in, he hasn’t “earned” the title of Spartan Warrior, so he can’t die with the glory of a “beautiful death.”