That silver wire plays three roles.
1. As a support wire or guy wire for the other two wires.
2. As the neutral or return path back to the utility pole transformer to complete the circuit for 120 volt power.
3. To electrically tie together (bond) the ground rod and/or water pipe ground in your house to the various utility pole ground rods and overhead ground wire that ties together the latter.
With the broken connection you had pictured, role #2 was officially wiped out and and role #3 was probably impaired. The clamp on the pole holding the silver wire just before the broken end performs role #1 and may have performed role #3 to a limited extent.
Given the way the system is assembled, role #3 allows a small amount of role #2 to be performed but the amount is unpredictable and it is a safety hazard. You may have gotten away with having some 120 volt equipment such as refrigerator, some components of your furnace, and lights still working correctly. But the more 120 volt usage you have, the more likely you would run into problems.
240 volt usage does not rely on the silver wire (it uses the two black wires only) but most 240 volt appliances have some 120 volt subcomponents in them. The actual current that role #2 is supporting at any given time is the difference between the 120 volt usage (amperes) drawn from one black wire and the 120 volt usage drawn from the other black wire.
The earth (between your water pipe ground or ground rod and the utility pole ground rod) performs role #3 and a little of role #2 but to a very limited extend because earth is not that great a conductor. (The silver wire performs the role much better.) Had the silver wire come completely loose from your house and laid on the ground, you might have gotten away with two or three amperes worth of 120 volt usage (specifically the difference between 120 volt usage from one black wire and from the other black wire). More than that and lights, etc. drawing from one black wire will receive more than 120 volts and lights, etc. drawing from the other black wire will receive less than 120 volts where the total comes out to 240. This imbalance can seesaw back and forth as you turn things on and off.
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