Is it permissible to subscribe to optional social security to provide for the needs of individuals, especially in old age and inability to work, given the state's failure to care for them, and does this contradict reliance upon Allah?

Social security that is Islamically permissible is not blameworthy. As for social security that resembles commercial insurance, it is forbidden, and the state's negligence does not justify entering into it. Entering into forbidden contracts without necessity contradicts reliance on Allah and blemishes faith. Allah has commanded piety and promised sustenance to whoever fears Him. So, seek provision from Allah, worship Him, and be grateful to Him. Fear Allah and be moderate in seeking sustenance, for no soul shall die until it has received its full provision.