I know there's the doctor in GTW EP but it's not what I'd like to see: it's focused on the career and not as much on actual sims (and, as an active career in a closed world, not my cup of tea). What I'd like to see is some kind of health system that would not be tied to a specific career and would give a major dose of realism to the game. And, as I'll point out, there's some base for it in the game already, it would just need to be expanded upon - to the size of an EP.

Injuries
There are injury-style debuffs that can happen when repairing stuff, the cramps/chills from IL patch, insect bites and campfire burns from OR GP... and they are all pretty much meaningless.
What a pity.
OR also gives some hint of possible treatment via herbalism (which could get some attention) These small things could add depth by needing treatment, such as band-aids for the cuts, herbalism/store-bought salves for bug bites and small burns, painkillers (with possible negative impact from overuse). Furthermore, pushing a sim too far with low fitness skills could result in stronger/longer cramps with different ways to solve them - massage or painkillers. Long dance sessions could lead to cramps/injuries (and could be a way to bring down sims' obsession with stereos). Another idea is low-skill cooks cutting themselves.

Sicknesses
These would have much larger potential with Seasons pack - being outside in rain/cold/hot weather while being inappropriately dressed would bear the risk of getting cold or hyperthermia. Sunburns could be treated over time with salves. Flu in autumn/winter as a contagious disease, sick days in work, and other realistic elements could be introduced (being drenched in the cold leading to having a cold, staying at the red part of hygiene bar could cause sickness, ...). Overeating, eating spoiled food, or using poor-quality ingredients could lead to stomach cramps and/or digestive issues. Maybe migraine from work could be another realistic element (impacted by traits, more on that later).

Allergies
Pretty self-explanatory. A sim could be allergic to pets with the EP installed (cat, dog, or both), flower pollen (bound to the active season with the EP installed), insect stings (bee box from seasons, OR insect bites). Food intolerance (nuts, milk) might be another option (and would lead to 'nut-free' and 'milk-free' tags in the appropriate foods, similar to vegetarian-safe we have now. (And, by extension, the health system EP could add hazelnut trees).

Skill impacts
Obviously, higher fitness skill would gradually reduce the chance of injury/cramps, as would higher handiness skill. Wellness and herbalism skills would get some use, as I hinted - relieving cramps through massages, herb tea for small ailments, salves for burns, etc.

Gameplay impacts
I've said that there's minimal impact from the negative moods caused by burns, etc. Let's take cramps, for example. The buff is there, and that's it. What I'd love to see is for it to start with a light cramp, giving you a time to get out of the water (if swimming) or stop the workout. If you did not, it'd go into a strong cramp, which would force stop you and prevent further physical activities (and potentially drown swimming sims). Likewise, a sim treated with cuts/pinched finger/electrocution from repairing/crafting would be prevented from returning to the activity.

Pharmacy and medicine
Medicine could be obtained either by ordering over the internet (computer/phone) or by traveling to a pharmacy (which players with GTW could run). This would include allergy pills, painkillers, first-aid kits (band-aid for cuts), pregnancy tests (which would need to be bought and then used instead of just clicking the toilet), burn treatments, etc.

Health trait slot
What I'd also include is the fourth trait slot (maybe two slots), which could only be used for traits specifying the health state of a sim. In fact, one could be player-chosen and one passive, the passive one based on the current state of a sim (fat sims being more likely to be ill, pregnant sims more likely to have digestive issues, elderly sims more prone to cramps). This would lead to several new traits for this slot to define sims with a specific strength/weakness (allergies are one example) and changes to some existing traits (some of which would be blocked by 'normal' traits that have the opposite effect - see below).

Other new traits
Handy: lower risk of injury when repairing/crafting. With OR, lower risk of being burned by fire. Faster handiness skilling by a bit, higher quality crafts. With Spa day, lower fail rate on giving massages (on lower skill levels) and/or possibly longer duration of the massage effect (25% or something).

Existing trait added mechanics
Base traits
Active: less likely to be injured/get cramps when exercising. Higher chance of becoming sick after a long period of no physical activity (the 'needs exercise' debuff would kick in at a set time such as 2-3 days, which would also trigger this higher risk).
Ambitious: immunity to headache from work
Child of the Ocean: immunity to cramps from swimming regardless of skill level
Clumsy: higher risk of injury but lower negative reaction to it (debuff one point weaker so uncomfortable +2 would be just +1 for them)
Family-oriented: sad when a relative is sick
Foodie: 100% chance to get sick on low-quality or spoiled food
Glutton, Slob: immunity to food-related sicknesses
Hates children: negative reaction to children doubled if the child is sick
Hot-headed: becomes angry when injured, higher risk of cramps during an anger-powered workout
Lazy: more likely to get injured from any active source (crafting, repairs, workout)
Loves outdoors: immune to plant and insect allergies
Loves cats/dogs: immune to allergy for that pet type
Mean: can mock sick sims
Outgoing: lower chance to get sick from interacting with others but the 'perk' is disabled after a long period of no interaction outside of the household (similar to the 'active' trait, think training the immunity system by being exposed to others)
Squeamish: becomes uncomfortable (3 to 5 based on severity) when injured/sick or (1 to 3 based on severity) when someone injured/sick is nearby
Vegetarian: the negative reaction to eating meat could result in the need to vomit or something
Reward traits
Responsible (Parenthood GP): same as Ambitious
Irresponsible (Parenthood GP): same as Lazy
Insensitive (Parenthood GP): same as Mean
Compassionate (Parenthood GP): some kind of 'it'll be okay' interaction on sick sims.
Master of the Sea (IL/Conservationist reward): same as Child of the Ocean
Long-lived (Athletic aspiration): complete immunity to sport-based injuries, can work out longer before being drained as an elder
Potion master (Mixology aspiration): probably some 'get well soon' drink that'd reduce the duration of any sickness by a bit (only once per being sick)
Handy (Nerd brain aspiration): immune to injuries from upgrading/woodworking
Naturalist and Survivalist (gardening/OR aspiration respectively): immunity to insect sting and pollen allergies
Seasons satisfaction traits: higher resistance (lower tier)/immunity (higher tier) to sickness caused by hot/warm weather
Waterproof satisfaction trait: lower chance to get cold from being soaked (complete immunity with Storm Chaser).
Frugal: 20% discount on medicine
Super Green Thumb: same as Naturalist/Survivalist

Cross-pack features
All the above shows major synergy with Outdoor Retreat and Seasons but is not limited to those. GTW could run a pharmacy and there would be a way to combine this with the doctor profession. Mermaids could have baseline immunity to swimming cramps and getting a cold from being wet. Animal allergy could prevent being a vet. There could be new massages for Wellness skill with the sole purpose of relieving cramps. Sims with pollen allergy could have a negative reaction to being given a flower. Beachcombing from IL could be a source of cuts from creatures lurking in the sand. Wearing dirty clothes (Laundry SP) could be another potential trigger for a sickness - and washing laundry used by a sick sim might need longer.