Earlier quoted context omitted.
Men under the age of 26, documented or not, are required to sign up after being in the US for 30 days, unless they are on a valid visa. https://www.sss.gov/register/immigrants/
That's murky and it's not enforced. In practice, immigrants aren't conscripted during a draft. If you want some more examples: Voting, Jury Service, Employment Authorization, and Driver's licenses.
In practice, immigrants aren't conscripted during a draft.
We haven't had a draft since the 1970s. It's unclear how many draftees were immigrants back then (and the legal definitions around immigration have changed in the meantime) but historically conscription of immigrants has been the norm, not the exception.
Voting, Jury Service, Employment Authorization, and Driver's licenses.
These seem like things from which undocumented immigrants are excluded (by law!), not immunized. Why you think this is an argument in favor of your legal theory is beyond me.