In 5e, a character is not even visibly wounded until they’re at half-HP. And at that point they’re not necessarily seriously injured, they’re just visibly battered, bruised, and bloody. It’s not until hitting 0 HP that a character takes a potentially life-threatening wound.
If that paradigm of HP breaks your immersion, I understand. Personally, the idea that a human(oid) can take six arrows to the chest and survive to even take an 8-hour rest is far more immersion-breaking than the idea that those six successful longbow attacks were not actually direct hits, but narrow grazes and glancing blows that left you harried and worn down but not injured badly enough that a night’s rest won’t have you back on your feet. But to each their own.