Lehrbuch
First Post
Tool proficiency is the kind of awkward side system of skills. In general, most of them seem to be fluff/ribbons, but then you have Thieve's Tools, which reigns supreme among them in most campaigns in terms of usefulness.
Yes, but in-play if a PC has a tool proficiency, then the player generally finds all sorts of pretexts to use it. At face value, Proficiency in Weaver's Tools doesn't sound especially useful. But if a PC has it, then most players will try to find creative ways to use it, which usually leads to good opportunities to earn Inspiration and just generally progress in the adventure.
Also, Disguise Kit, Forgery Kit, Herbalism Kit, Navigator's Tools, Poisoner's Kit, and Vehicle Proficiency are all tools that are pretty generically useful, like Thieves' Tools without the player needing to think very hard.
Like how do you know how to make all these things with an herbalism kit, but have no knowledge of herbs, which is covered under Nature?
If you actually read the rules you will find that "Proficiency with this [Herbalism] kit lets you add your proficiency bonus to any ability checks you make to identify or apply herbs." (Basic PHB, 51).
You could probably also use Intelligence (Nature) to identify and find herbs, but probably not to apply them.
Also, you seem to be forgetting any character can use every skill, tool, etc. It is just that characters are only proficient in a limited number. Nonetheless a PC with a good relevant Ability Score will still be quite good at a skill they are not proficient in.