The optimal betting strategy would be to "invest" a fractional percentage of your bankroll. See the kelly criterion from above.
Not doing so is suboptimal
Sure. But you said "This is not true at all and a common fallacy". In fact, it is quite possible to reliably win when the odds offered are better than the actual odds.
I guess we're just talking over each other - you made an assumption that I don't believe was clear from the context.