Okay, so this one is a system, but it will be heavily restricted in its abilities. Plus, this is a Toriko idea, and it has plenty of content to work with. This is intended as a reincarnation idea, but it could easily be reworked for either transmigration or an insert. Let's start off with the system's description.
So, the Gourmand System is designed entirely around eating, as its currency, GP or Gourmand Points are earned through consumption of various ingredients. GP is only earned through the first time eating an ingredient, though the MC can earn more through eating ingredients different ways. I'll give an example: An egg gives 10 GP, black pepper 5, and salt 5. The protagonist will earn 20 points consuming these separate, but will also earn another 20 if they are cooked into a dish, like scrambled eggs. The more rare and delicious the ingredient, the more GP is earned. Now, what can GP be used for?
Gourmand Points are one of the ways the MC can increase his strength. GP can be exchanged to increase strength, endurance, agility, as well as his five senses, sight, hearing, smell, taste, and touch. That's not to say he can't grow stronger through training. Later on, it can also upgrade his Gourmet Cells, which will strengthen his Appetite Demon as well. Another one of the Gourmand Points uses is to instantly learn or improve a skill. What they can't do is buy things, as there will be no shop, though if desired you could include an inventory. Now, about how to visualize the MC's stats.
I'm not entirely sure about later on in the story, but at the start you could visualize it like this: All his stats start off at 1, which would represent the average man. He would have the strength of one man, then two men after an upgrade, and so on. Maybe you could do it according to Capture Levels later on in the story, since they seem to stay pretty relevant. It's kind of hard to gauge if Gourmet Hunters can be held to that standard, but I'll just assume that they can.
If you so desire, I can suggest a few possible romance options, or harem members if you're into that. Well, actually, there's really only one, though it may just be my bias towards this character: Melk the Second. After looking at the wiki a second time, there is the movie character Ayame, a head chef of one of the Biotopes. If you want more, you'll probably just have to come up with new female characters.
I suppose that it might be necessary to include a possible martial art for the MC, so I'll list two suggestions: Kyotoryu and Hiten-Mitsurugi Ryu. Both are sword styles, one is sword-less, the other is all about speed. I much prefer Kyotoryu, since it doesn't require a sword, but since I lined up Melk as a possible love interest, perhaps Hiten-Mitsurugi has a place as well.