The Pilgrims landed in 1620, the year generally accepted as the dawn of the true flintlock, which was the standard lock mechanism for the next 200 years.
Wheellocks like this example had been existed since the mid-1500’s. Its great advantage was to eliminate the need for a constantly burning `match’. The wheellock had faster lock time than the flintlock that succeeded it (meaning flash-to-bang time) but was much more complicated.
Matchlocks remained in use well into the 20th century among Afghans & other Central Asian peoples.
The gun mistakenly used to symbolize Pilgrim arms was the blunderbuss, a weapon meant for defense at close range with its bell mouth muzzle, and nonexistent in the early 17th century.
Would that have been what they referred to as a “fouling piece”?