I have never of a PUSHBUTTON connector.
Now there are some push on connectors and Lowes has them. But as far as I know they are not reversable. To get off or to remove the old fixture you need to cut the wires.
If you have that type and tried to pull the wires out then you probably have damaged them.
Use wire nuts and eliminate that possibility.
Because they are now reversable and have to be cut off I only use them where the wires are too short. Then I connect them to pigtails which I then connect to the fixture or device.
And yes switches that are starting to fail can be intermident.
Aslo I have see sockets in light fixtures with bad connections.