Is Weakness Necessary?
In that sense? No. I cannot recall- nor would I like to- a time where humans preyed on each other regularly for direct sustenance- in terms of literally eating one another. As far as conquering land and peoples for resources or servitude, yes but even that doesn't mesh with the reasoning in your OP. Animals don't enslave other animals, they eat them- or at the greatest congruence to humans, chase them out of former territory.
Humanity however is a bit different. When most people think of the word 'weak' they think of lacking physical strength. Seldom does mental ability come to mind. And sure, living as cave peoples someone bigger than yourself makes you for all intents and purposes- weak. As in your power and potential over theirs is far lesser. At least in comparison. But- if you were to say utilize something as simple as a lever and wield something as simple as a slingshot for example, that advantage in strength and size the other person has over you is negated. Or at least, the two are drawn much closer together. Sure, greater physical strength and size will always translate to greater power in terms of analog devices and tools but it becomes less of an inherent necessity.
If you're from one group who is being invaded by a much larger group, with few numbers left on your own- and you manage to study natural elements and determine which are poisonous- you can use stealth to say "spike" the enemy camps water supply and become victorious by the next morning. Just one of many examples.
Say you're in a primitive jungle and you come across someone much larger than you. If you can either get them to chase after you or perhaps chastise them if they don't at first, you could lure them to say a pitfall trap with sharpened spikes. Or just in general take someone much stronger than you. If that person is vulnerable to name-calling, insults, and other forms of mental sabotage and you happen not to be, well, you have the upper hand in most things save for a direct conflict of course. And even so, anger or rage especially can lead to mistakes and oversight.
To answer the question, assuming we're using weakness and strength properly and not completely focused on the physical... generally probably not but I'm sure there are some exceptions. I'm trying to think of a more eloquent way to phrase it other than "don't be like that guy" in terms of motivation to become the strongest you can be, especially if you witness reasons or events where it would've come in handy. Besides, many people and I would hope most train physical strength because they know it's smarter than to not and perhaps even to do good things, but those who don't and do so just for vanity or rather to try and address a deep-rooted personal insecurity or inferiority complex... would probably feel better about themselves after coming across someone less gifted or otherwise "weaker". I suppose there's that.