My thoughts as to why Eurylochus was second-in-command and not Polites
First of all, the danger aspect. There is a good chance that the second-in-command has to be on the front lines with the captain because they’re the person the crew looks to if the captain isn’t present. Thus, Odysseus didn’t make Polites his SIC because he didn’t want to put Polites on the front lines. Hence why Polites is less cold and aggressive than the rest of the crew; Odysseus preserved Polites’s innocence, in a sense, by keeping Polites off of the front.
Second of all, connection with the crew. Both Eurylochus and Polites seem to be the kind of people that approach others first and care about them. However, with an army of soldiers who are hardened for battle, Polites might not be someone they would trust with their life because Polites is working with an ideology that is at odds with their own lived experience. He’s more gentle and assumes the best out of everyone, saying that putting your best foot forward will encourage others to do the same. That isn’t going to keep the crew alive in war. Polites is also less strong in sharing his ideas. When pointing out the island of Lotus Eaters, his idea is just “maybe they’re friendly and willing to share food with us, who knows”. He doesn’t want to hurt anyone and isn’t very certain about his solution to the crew’s problem of food. He even puts in “maybe” and “who knows”.
Eurylochus, however, is more assertive. He always initiates conversations with Odysseus to make Odysseus aware of what is happening with the crew and about his opinions. In FSP, Eurylochus is the one that makes Odysseus aware the crew is starving. Eurylochus doesn’t say “maybe we should strike”, he says “I say we should strike first, we don’t have time to wait”. Aka more certain and strong in his ideas. In Storm, Eurylochus is the one that tells Odysseus about the concern that the crew will not survive trying to push through the weather. In Polyphemus, Euryochus immediately focuses on how the whole fleet can be fed with all this food and is relieved while Polites is just thinking about how much they can keep. In Luck Runs Out, Eurylochus fights back on Odysseus’s “plan” out of concern for his captain. With Eurylochus being much more concerned about the needs of the crew and certain in his ideas and standing up to protect others, the crew is more likely to trust that Eurylochus has their back.
Third, their interactions in Puppeteer and Scylla. While unfortunately we couldn’t see much of Polites, we do get to see how Eurylochus treats Odysseus. In Puppeteer, after Poseidon kills the crew saying that it is because of Odysseus’s actions with Polyphemus that he is here to get vengeance, Eurylochus solely focuses on his accountability in the interactions. He doesn’t try to blame Odysseus for pissing off the God in the first place, nor does he try to say the Winions or the rest of the crew goaded him into it, nor does he try to hide away until Odysseus brings it up or something. Eurylochus is the one that goes up to Odysseus first, without any sort of prompting, to try to confess that he made a mistake that cost the crew. He then stops when he notices Odysseus isn’t all there in the moment and tries to persist when Odysseus orders him to start scouting out the island, only to finally concede when Odysseus says it can wait.
Not only that, but when Eurylochus is describing how the men fell for Circe’s tricks, he doesn’t go “oh, well they fell for it and I didn’t because I could just feel something was off”. He says “But this is a hell of a twist, ‘cause we are weak to a power like this… She had us inside in just two words.” Eurylochus is very open about how he was weak to Circe, too. So I headcanon that the reason why he didn’t go inside even though he admits he fell for the trick as well is because he felt guilty of betraying Odysseus in a moment of weakness and resolved to stay outside to be dutiful while letting the rest of the men go in to rest up because he feels guilty for murdering most of the crew. Only to then watch in horror as they were turned to pigs and run back to save Odysseus and the rest of the men from this cruel temptress.
In Scylla, Eurylochus once again reaches out and tries to get Odysseus to open up. And after getting pushed out by Odysseus with a refusal to be vulnerable, Eurylochus decides that there will never be a better time with Odysseus pushing him away and confesses his shame. Being vulnerable with Odysseus after Odysseus refused to be vulnerable.
From those two interactions, I feel like Eurylochus had been the one to make this “friendship” with Odysseus. Because throughout the whole musical, Odysseus is not an initiator. He hides away, he bottles up his real feelings, and he is very independent because he thinks “I’m the leader and they follow me and they rely on me”. And Odysseus is not close with any other crew member. Meanwhile, Eurylochus makes sure to be aware of the crew as living beings with needs, which requires speaking with them and micromanaging them. Eurylochus notices when Odysseus is down and makes moves to help. But Odysseus never does ask what anyone else is feeling. He never notices anyone else is sad or tries to help comfort anyone else. So when it comes down to it, it’s likely that Eurylochus bonded with the crew more personally and then bonded with Odysseus while Odysseus never tried to make any personal connection to any of the crew.
Which is why Odysseus took the betrayal of the wind bag so personally. Because he thought he could rely on Eurylochus to always have his back just like everyone else in the crew because Eurylochus is responsible and caring and had been the one to take that step to become friends with Odysseus (in a sense), only to learn that Eurylochus was the one to betray not only him with the bag but the rest of the crew as well by running away when Circe turned the men under his temporary command into pigs. With the only solution to just run and not look back.
So while Polites and Eurylochus are pretty similar in that they both want what’s best for the crew and specifically are the two that have some sort of friendship with Odysseus within the army (Odysseus only calls Polites and Eurylochus friend while everyone else is brother or comrade), I believe Odysseus felt he needed to protect Polites because they were best friends since childhood (at least from what I remember), but Polites refused to let Odysseus go to war without him. And when this man in Odysseus’s army (who is strong and responsible and assertive and trustworthy due to his honesty and created strong personal bonds with the army) comes up to Odysseus to make some sort of bond with Odysseus, Odysseus thought Eurylochus would suit the role of SIC perfectly.