I did some testing for these questions in game since I couldn't find any answers to your questions online.
There are two characters who can consistently take advantage of spirits that increase stats after eating: Peach and Daisy. Their final smashes spawn food that when consumed do indeed activate those abilities. This is easier to activate in Adventure Mode where with the Skill Tree, you can increase the rate at which the FS Meter is charged and/or filling the FS Meter gives two Final Smashes. Spirits like Ashley can also increase the rate at which the FS Meter is filled. Mr Game and Watch can also spawn food with their Side Special as you described, alas inconsistently.
This works both for spirits that increase stats after eating, and those that grant invincibility after eating - I tested with these spirits:
Screenshots of this happening with Daisy:
Inhaling/eating opponents with Kirby/Dedede/Wario doesn't count as eating and doesn't give stats. Only eating food does.
Otherwise characters that run fast like Sonic could theoretically take advantage of those spirits better since they can reach food faster. Similarly spirits that give "Item Gravitation" are also helpful.
Other observations I made:
- Eating food and Maxim Tomatoes activate these abilities but not
Hearts or Fairy Bottles
- Food, Maxim Tomatoes and Hearts cannot be inhaled with Kirby or Dedede but
Fairy Bottles can. Wario can eat all of them though
- Inhaling items like Barrels with Kirby/Dedede or eating them with Wario do not increase stats