You sound like you're overvaluing tranquil boots. It really doesn't do anything that a lich couldn't do without. Ring of protection gives you nothing except armor.Armour mitigates physical damage.A better way to achieve that is to chose your trades better so that you lose less hp than the other guy.Another way to to buy more regen like a salve.The RoP costs more than a salve,only covers you against physical damage and even then only makes you live for another 3-4 right clicks.Salve costs 65 gold less,gives like triple the effective hp and works against magic harass too. Sentries on the other hand stop heroes like treant, weaver, riki, bounty from playing at their full potential because they pose no threat if you can see them sneak up on you. If you see a ward you get to instantly deward that and get you full money back (you can buy 1 sentry now as of 7.02).Sentries are also useful to give to your mid if hes gonna be ganked by invis heroes and of the enemy has a highground ward for them.

TLDR Sentries give you something that is irreplaceable by any other item or by player skill while RoP does not. If you find me a purge video where he buys a RoP at the start then I'll delete everything I just typed to prove my point.