That socket can be replaced. It has two tiny screws from the back side to hold it in. The trick here is to get the right socket and get one that is pre-wired with lengths of wire to feed down the 3/8" flex whip to the junction box on the light.
I am looking for that same exact socket, in large numbers.
Good luck
The answer is not a cut and dry one in your case. The troubleshooting is a process. You will need an understanding of a multiple light parallel circuit. You will also need a proximity type voltage tester and possibly a VOM.
Assuming the voltage feed starts at the switch (not always the case, but normal) you will need to verify input voltage at the line side of the switch with the switch in the off position. If you have voltage there, turn the switch on and verify voltage on the load side of the switch. At this point, a VOM is handy to test voltage across the hot and neutral and/or ground. If this looks good, proceed to the closest light fixture, and with the switch on, test the center hot tab in each fixture with the no-touch tester. In your case I would think you may not see any voltage based on your question. This could mean that you have an open neutral, an open hot, or at worse, a shorted hot to neutral/ground.
Since there are several possible reasons for your condition, it would take a lot of tutorial to explain every possible scenario. Assuming you do not see voltage at the fixtures, the basic technique for troubleshooting will be to start at the last known verified voltage point, then follow the wiring and check all connections in the junction boxes. Visually check the bulb sockets for broken or shorted metal tabs. This should be done with the power off at first, looking for obvious loose or disconnected wires, then with the power on using your voltage tester. Since this condition occurred after you adjusted the height of the sockets in the fixtures, I suspect tension on some wire may have pulled a wire out of a socket base or out of a wirenut in the j-box. Unfortunately, most can/pot lights have a built in j-box attached to the top of the fixture. This necessitates dropping the fixture down below the ceiling to access the j-box, or getting access from above. (attic).
Basically, you are following the circuit looking for an open or short, just like following a hose, looking for a water leak or stoppage.
This is not a hard job, but extreme caution must be taken when testing energized circuits. If you do not have good electrical skills, the proper test equipment, or a logical understanding of switched paralleled circuits, then this job is better left to a pro.
Maybe one of my buddies here on SE can add a good graphic showing this type of circuit and the test points. A simple line drawing showing the switch and junction points would be a great edit.
Best Answer
Recessed can lights have several issues you have already checked the center contact, now there are 2 other things to check. First I would look for a thermal overload, these are a small sensor that is inline with the hot conductor to the center tab on the light socket, if a larger lamp was used than the fixture was rated for this is usually the cause, the overload is temp sensitive and opens when hot, these sensors don’t last long if a over sized lamp was used in the fixture they are usually small some about the size of a pea, many about an inch long and smaller than a pencil in diameter, to check this the circuit needs to be turned off and an ohm meter from center contact to the black wire should read close to zero if open the overload has failed and needs to be replaced (sometimes cheaper to replace the fixture. The other thing I find on older fixtures is the heat for all the years has baked the wiring and with some vibration one of the conductors opened at the fixture or the splice to the power that should be in a box ether on the fixture or connected close by.