Conscription is not considered grounds for granting asylum, not from any country. Israelis can't request asylum because they would be conscripted into the IDF either. Conscription is just the law of the land that applies to everyone in it, it doesn't target people specifically, regardless of any protected class like race, sexuality or political opinions.

Asylum would require something like political persecution, e.g. standing up to Putin and becoming a target of the FSB; that would be grounds for asylum.