I should say that in the case "I am tired nevertheless I work a lot" you better take "и всё же" or " но всё же". Simple "всё же" is closer to " I would admit", "I should say".
Я устал, всё же я много работаю - the sense is like "No suprise that I am tired as I work so much"
Я устал, но всё же я много работаю - I am tired but I continue working hard

When you put "но" - you talk about smth opposing the statement in the first clause - Сегодня дождь. Но всё же погода хорошая. "И" is somewhere close or even the same. And this you can translate as nevertheless. Simple "всё же" is proving your first statement, adding one more argument to it. Like in your sentence about "закалённые люди" - it is not so sharp as "since" because it is not a direct conclusion, but like you speaking with yourself, weighing pro and contra.