When you say neutral, do you mean his neutral-air? If so, it's more of a get-off-me move than anything else. Forward-air's probably a better aggressive move in most situations. The hitbox is bigger and it creates more stun on shields. The only real downside is that it's slower.
As a whole, Tink's game plan is usually to use projectiles to open people up so that when you go in with the sword you don't get bopped. His sword moves, as you said, aren't the greatest. They're not meant to be. If they were, he wouldn't need projectiles. He actually has better combos than the other Links, but he doesn't have much access to them because they're so specific. (Think up-tilt -> back-air -> back-air -> up-B. That does about 40% and works, but your opponent has to already be at about 50% depending on the character, and on top of that you usually have to be facing away from your opponent. They basically have to land on you or whiff something.) He does get some short combos off his boomerang and bomb, so you may want to practice those. I've seen a lot of people have luck using his aerial hookshot, too, so that might be a thought.
My personal experience is that Tink does well when he's able to limit the other person's options and play mind-games with them. If you throw a bomb and they shield, that doesn't mean you've lost the situation. A lot of times people will sit there in shield, so you have a free opportunity to do something. You could grab them as they sit in shield, catch the bomb as you hit their shield with a forward-air, or just walk away and let the bomb explode while you do something else. You just have to set up the situation and be ready for it to happen so you can take advantage of it when it does.
It'll take time, practice, and strategy to pull his potential out, so I hope you keep working at it. It was the same in Smash 4, where the general consensus was that Toon Link was mid-tier until the final year of the game's life, where people like Esam started placing him in high-tier (around 20th place) because the few people who were playing him were starting to get good results against some of the best characters in the game. Good luck, and I hope Tink becomes less frustrating for you.
