Only magically-created food has rules for spoilage
"Food scraps" are not really a thing RAW. The solitary reference to food scraps in the PHB is the Urchin Personality Traits table you referenced (p. 141). In the DMG, the only reference to "food scraps" is as result 51 on a random table of "General Features" in "Dungeon Dressing" (p. 299). Since there is no definition of food scraps as an item, we can assume that they are just bits and pieces of food.
RAW, food may be obtained by characters through purchase, though foraging, or through magic.
Purchase ("Adventuring Gear", PHB 153)
Rations. Rations consist of dry foods suitable for extended travel, including jerky, dried fruit, hardtack, and nuts.
Purchase ("Food, Drink, and Lodging", PHB 158)
The Food, Drink, and Lodging table gives prices for individual food items and a single night’s lodging. These prices are included in your total lifestyle expenses.
Foraging ("Foraging", DMG 111)
A foraging character finds nothing on a failed check. On a successful check, roll ld6 + the character's Wisdom modifier to determine how much food (in pounds) the character finds, then repeat the roll for water (in gallons).
Magic ("Create Food and Water", PHB 229)(emphasis mine)
You create 45 pounds of food and 30 gallons of water on the ground or in containers within range, enough to sustain up to fifteen humanoids or five steeds for 24 hours. The food is bland but nourishing, and spoils if uneaten after 24 hours. The water is clean and doesn’t go bad.
Notice that in all these sources of food, the only one that specifically mentions spoilage is the magically created food. Interestingly, at this point in the development of 5e, there is no rules mechanism by which normal food spoils.
The rations one can purchase for "extended travel" certainly seem like they are types of food chosen to avoid spoilage, but notice that there is no mechanism by which fresh food foraged will go bad - no condition that raw meat you foraged in the tropical heat won't be just as suitable to eat next week as it is now. Note that I am not saying that normal food doesn't spoil - just that rules for this, the rules the OP asked about, do not exist.
This is a clear departure from previous editions, in which one could cheaply purchase "standard rations" (which would quickly spoil) or could pay much more for "iron rations" (which would last much longer). If you are interested in the subject, you might want to start with this and this.
Furthermore, in 5e there is no mechanism by which spoiled food may be made wholesome again. The spell that previous editions used for that simply does not have that function.
"Purify Food and Drink", PHB 270
All nonmagical food and drink within a 5-foot-radius sphere centered on a point of your choice within range is purified and rendered free of poison and disease.
Compare that is previous versions of the spell, such as in 3.5e:
This spell makes spoiled, rotten, poisonous, or otherwise contaminated food and water pure and suitable for eating and drinking. This spell does not prevent subsequent natural decay or spoilage.
The first answer to your question is that 5e simply has no mechanism for non-magical food spoilage; the scraps you have in your pocket, even uncooked meat scraps kept at room temperature, will not spoil as a consequence of anything written in to the game. If it is a truism that D&D is not a Physics Simulator, in this case neither is it a Biology Simulator. If this is a level of detail you or your DM would like to incorporate you can do so. This could simply be by DM fiat, telling you that you scraps have spoiled, or you can create a more formal set of house rules. You might find previous editions useful in this regard. Working within the mechanics of 5e, the simplest thing to do might be to come up with a table for how long different kinds of foods may be kept before they are considered "poisoned".
So what is the point of the personality trait?
As you say, you have so far just been collecting these scraps for role-playing your personality trait, it it is fine to have it remain there. The purpose of having specific, listed personality traits is exactly for this type of role play:
"Personality Traits", PHB 123
Give your character two personality traits. Personality traits are small, simple ways to help you set your character apart from every other character. Your
personality traits should tell you something interesting and fun about your character. They should be self-descriptions that are specific about what makes your character stand out.
However, personality traits do not have to be limited to role play - they can have a mechanical effect, through Inspiration.
"Inspiration", PHB 126:
Inspiration is a rule the Dungeon Master can use to reward you for playing your character in a way that’s true to his or her personality traits, ideal, bond, and flaw. By using inspiration, you can draw on your personality trait of compassion for the downtrodden to give you an edge in negotiating with the Beggar Prince...
Your DM can choose to give you inspiration for a variety of reasons. Typically, DM's award it when you play out your personality traits, give in to the drawbacks
presented by a flaw or bond, and otherwise portray your character in a compelling way.
Since you have been role playing this all along, you may discuss with your DM whether they will be awarding Inspiration at some point based on your role play. They could use this to give you a bonus in a certain situation - as other posts have suggested, you might get Inspiration to use on an animal handling roll if you fish some table scraps out of your pocket before your interaction. Or if you are at a fancy dinner and insist on lowering your reputation with the host and other guests by saving food in your pockets, your DM might award you Inspiration for staying true to your personality.
None of this use of your personality trait needs to involve the food actually spoiling or the need to introduce new mechanics, since it is entirely up to DM fiat.