Savings in an economy can comprise of actual investments, and unspent income. Any unspent income does not contribute towards the GDP, so it can't be part of C+I+G+X-M. Any unspent income will actually lead to lower GDP in the next time period.
If I means income spent on capital investment, and savings includes unspent income, why will S=I? Savings a.k.a unspent income doesn't lead to more capital investment because, even if deposited in banks, it doesn't lead to more lending for investment. After all, banks don't need deposits prior to lending. It's lending that creates deposits. Also, savers don't normally save to put to investments later on. Many just want to have a cash buffer for consumption, if they foresee not having enough coming from income in the coming time periods.
So why keep S=I? Investment may include inventory, which may have remained unsold by businesses during the current time period. But unsold inventory usually isn't a result of businesses wanting to spend more of their income on more unsold inventory. They're usually a consequence of buyers deciding not to buy. So an increase in household savings does not necessarily mean an increase in business savings. It may actually mean a decrease in savings by businesses because businesses may now have to liquidate those unsold inventory at a huge loss, and not necessarily by selling to households at discounted prices. i.e., increased savings for households. Some unsold inventory may simply go to waste, the scrap heap, meaning, decreased savings for business (they'd used previous income to purchase the inventory) and decreased business income for the current time period.
So why keep S=I? It assumes the economy is in equilibrium, where all savings goes towards investment, which it never is and never does. Nowadays, I can constitute not just of S, but increasingly, of B (Borrowing). This B doesn't come from S because again, banks don't need deposits prior to lending. It's lending that creates deposits.
Any unspent income, not put into actual capital investment, for all intents and purposes, has now exited the economy (unless it's used to pay off their B, which doesn't really contribute to GDP, but can be considered as decreasing banks' I).
