My take, for what it is worth!
Tessio was protecting his territories rather than being "an ambitious snake"

Tessio should have had more faith in his Don/s and trusted their judgement

I believe, Tessio only became a "snake" because Tessio felt Michael would be no good as Don even though Michael kept reassuring including -
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Be patient There are things being negotiated now that are gonna solve all your problems and answer all your questions

always smarter! Tessio jumped what he thought was the Corleones' sinking ship
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Barzini's people chisel my territory and we do nothing about it
Pretty soon there won't be one place in Brooklyn I can hang my hat!