Rest assured that your questions are very good ones
You are right if you only consider the relationship between companies’ capitalisation and the value of the SP500 from a static situation. i.e. if you freeze time and all prices and then mathematically work it out.
But markets are dynamic and share prices are constantly moving. Therefore one can consider the relationship between the cumulative value of the various companies and the value of the SP500 index to be something like a rubber band where one pulls the other.
But like the proverbial chicken and egg situation, which move comes first! The reason for the dynamic nature of markets in general is because speculative participants believe that price should be somewhere other than where it is right now. This same principle applies also to the movement of indices. Although they are based on the value of the underlying shares, if the value of the shares is expected to be likely to move then the index value will also tend to move towards that foreseen future value of the shares rather than where they are right now.
It may be that computer models are constantly and continually recalculating the value of the index as each individual component share price moves, but the index would surely pre-empt the move in individual shares, especially when some fundamental factor is seen to affect the overall stock market such as interest rate changes, inflation, fiscal stimuli, etc.
In theory, one could argue that if the index is out of synch with the weighted value of the actual component company values then arbitrage trading would bring them back in line with each other, and I guess at some point that would occur. But the physical process of actually buying and selling a portfolio of actual shares to mirror the index and benefit from a small incremental arbitrage gain would surely be expensive and somewhat inaccurate in constantly moving markets.
Index trading is not my speciality at all and I may be totally wrong here and it may be that other members here like @krugman25 can answer you much better. But in principle, the elasticity that is generated by moving markets will allow flexibility for the indices to behave, to a greater or lesser extent, as though they were actually a commodity themselves and not just a mathematical formula.