In my games, a set of alchemist's tools or herbalism kit is required to craft potions, but you have to be a spellcaster to make potions with alchemy. However, alchemist's supplies can be used to make a greater variety of potions, while herbalism is restricted to healing and protecting.
I find that this seems like the most logical way of going about it, other than the spellcaster only for alchemists. Alchemists don't necessarily need magical knowledge, I would extend this to anyone with the Sage or Guild Artisan who picks his trade to be alchemy, despite most people going sage background are likely spellcasters.
Most tools other than Disguise Kit, Thieves Tools, Smithing tools, Leatherworking tools, Herb Kits, and Forgery Kit are for flavor reasons. There is little need in most (non-social) campaigns for other tools, there are a few exceptions I can think of though.
Your DM allows players to create magical items in this manner, you would have to craft the base for the item whether it be a robe or etc. and then you have it enchanted to become a magical object. But this isn't that common in campaigns.
Carpentry tools or Masonry tools will be used for the 40gp cart you don't feel like buying for 2gp more from buying for a reliable merchant.
Some tools can be used to make situations more character focused and can possibly turn the situation in his/her favor, my DM will drop the difficulty of checks if your explanation is good enough. Just murdered a person that insulted your evil monk with weaver's tools? When the town's guard approaches just explain you tripped and fell on top of him with your scissors (two daggers) which broke in the accident, you'll still have to make a check but you took your murder's high check and turned it into a more manageable roll.
Tools such as cooking tools don't seem applicable at all unless you can talk your DM to apply benefits such as better food allocation, you don't need as much food to feed a character an acceptable meal, or in the case that you will need to personally entertain nobles.