Nah, making Blink un-Rearmable would completely destroy the hero. The reasons items like Midas and BKB are un-Rearmable is because they would be absolutely broken.
BKB - Spell Immunity forever
Midas - higher GPM than Alchemist
Necros - Instantly resummon every time they die, or about to die, and push another lane. March can't hit the buildings, and Tinker isn't very good at hitting the buildings. Having Necros always at your beck and call would make this super broken.
Arcane Boots - Infinite mana without any kind of cost (like Soul Ring), mainly this is a concern for the teammates. Just imagine a Tinker just constantly Rearming Arcane Boots and keeping his whole team's mana full all the time. None of them have to manage mana ever.
Refresher - Loophole for any of the above.
Making Blink un-Rearmable would be like adding a cooldown to Storm's Ball Lightning, or making Pudge silence himself for 3 seconds after hooking, or other things like that. Blink is absolutely fair, especially since the damage cancel is a thing. Tinker is all about being able to Rearm items, otherwise he would be completely useless. Why not say him being able to TP everywhere is OP, so make BoTs un-Rearmable? Scythe of Vyse? Dagon? It's all a natural part of the hero's kit.
Now what I would agree with (and I was surprised they didn't go this route back in that patch where he was OP), is having better on-damage detection for the Blink. Example would be if you were standing in some kind of DoT AoE (or multiple things hitting you), you Rearm and Blink instantly and get it because you re-Armed the Blink cancel CD.
He already got the damage type nerf with March, where the damage types were simplified, so Composite and Mixed were gone, and March was magical now and no longer went through BKB, and of course no more ancients (although it still aggros them and this has been a bug ever since the nerf, but they don't seem to care to fix it).
