He fades badly lategame, his damage is completely nullified by a bkb (he can't even attack while his chakram is out), and he has no disable. He only seems invulnerable when he is allowed to disengage and regen, or when it's the laning phase and he has some points in armor to completely nullify harass. In a teamfight the regen is never really a factor, and the +16 armor only kicks in after 16 physical attacks, which is frequently enough to kill him anyways. Basically, he has a few specific scenarios where his mobility, aoe pure dmg, and passive can make him a beast, but if he doesn't have a good angle to chain through trees, or you stay spread out so his chakram can only tag one, or you focus him down right after he chains, he's not as intimidating as he seems.
Timbersaw is one of my favorite and most played heroes, and there are definitely times where I feel invulnerable. Those times mostly occur because he needs to be approached differently than other heroes, where you have to strongly consider the surrounding trees and consider that his chain is a 4 second cooldown. It's hilarious when a chain around through 3 heroes trying to gank me and they don't time their disables. Getting stunned or hexed right after I chain is fine, because I'll just build up armor stacks and once the disable wears off I'm chaining off the screen. Unless you know you can burn him down before a disable wears off, it's usually better to anticipate his chain, nuke him to force the chain without building up his regen, then burn him down and disable him a few seconds later when his chain is coming off cooldown. There's nothing worse than initiating with a stun and physical dps, then having him chain away and regen up hundreds of hp while you chase him around the forest. Prolonged fights without heavy focused spell damage are Timbersaw's bread and butter.
I do think his spells cost too little mana, considering they all have some sort of utility (stat steal, mobility, slow, aoe) as well as pure damage (or at least the potential for pure damage). I rarely have mana issues before I'm buying arcane boots, and from then on you'll rarely get low mana again unless there is a prolonged fight. The passive is probably a bit too strong in the early / mid game, but it tapers off so badly later that I don't think it's imbalanced.
When I offlane I generally go armor / whirl / armor / chain, or some variant of those. Sometimes I'll have 2 in chain by level 4, but generally 2 in armor early makes you immune to harass as long as you steal creep aggro every now and then to keep your stacks at 8. Against two nukers, however, or a lane with strong disables, Timbersaw is pretty fragile still.