Yes, it is a problem with all the non-tilt aftermarket turn signal switches.
This problem is more prevalent on Cougars with 6 brake lights. Cars with only 2 brake lights (67-70 Mustangs) not as much.
Cars with 4 brake lights, like the 1971-73 Mustangs are more susceptible.
Since NOS switches are mostly gone, that may be your only option. The better switches have copper rivets - not silver shiny steel rivets.
Inspect the connections to make sure there are no loose rivets.
Operate the new switch 10 times in each direction before use (recommendation from an old Ford TSB).
Here is one way to test your new switch with a digital ohm-meter on the lowest resistance setting.
Measure the resistance across the pins with the switch disconnected:
Center position (brakes): green/orange, orange/blue, green - should be less than .1 ohm between any two wires
Right turn: orange/blue, blue (use the smaller gauge blue, not the 14ga horn wire)
Left turn: green/orange, blue
My first experience, in1976, was when an NOS switch released the magic smoke one week after I was installed it!
When I was daily driving my car, I replaced the turn signal switch about once per year. So NOS isn't necessarily better.
I finally disassembled a bad switch, made custom larger contacts, and used screws to reassemble it.
Unfortunately, this is not a cost effective solution.
The soldered rivet contact solution, from rmontoya746, is another possibility.
I would use a low temperature solder to reduce the damage from melting the surrounding plastic
