Is there a weighted shirt or similar which one could use for everyday
wear? ... Is there a more discrete option for this kind of training?
The most popular and simultaneously most discrete option for this kind of training is using ankle weights. These can be hidden, or at least mostly obscured under pant legs. And naturally people should be more focused on eye contact than at feet, so it is outside the normal focus of vision as well (unlike a weighted vest).
I'm hoping my natural daily activity can replace a workout routine
if I am wearing weights
No, this is not possible. While this depends on what you define as "natural daily activity" and also what your goals are (muscle, tone, strength, fat loss, health/wellness/longevity). Overall, there is absolutely no comparing daily activity with working out. Why not?
- Most Western daily activity is sedentary, and even working out cannot offset the damages of prolonged sitting.
- Daily activity does not raise your heart rate or involve anything remotely equivalent to cardio for aerobic conditioning.
- Daily activity does not stimulate muscle growth past their point of adaptation.
- If daily activity made people fit, then Westerners wouldn't be in the middle of a massive obesity epidemic (granted, diet has an equal if not larger share of guilt here).
If you want to put on muscle, you need to workout and eat healthy. If you want to burn fat, you need to workout and eat healthy. If you want to be strong, you need to workout and eat healthy. Are you seeing a pattern here?