Before the 1910s most banjos were strung with gut strings and played classic style - that is finger style similar to classical guitar and with no finger picks. This style was immensely popular and although there are quite a few players still today, it is mostly forgotten or ignored. Before that, there was a style called minstrel banjo which would eventually evolve into claw hammer / frailing styles, but that's another story.
In the 1880s and 1890s mandolins and mandolin orchestras were quite popular (as were finger style banjo orchestras). At the time these were about the only instrument to use flat picks and metal strings.
Musicians and instrument makers began to experiment with metal strings on banjos (and also guitars at around the same time). The most successful (or at least lasting) of these were Banjo Mandolins (aka Banjolins). But also there were (and still are) Zither Banjos - which are meant to be strung with a mix of metal and gut strings. (To string them with all metal strings can cause some of them irreparable damage). And the tenor banjo (which like the mandolin is tuned in 5ths). It began to make an appearance by about 1910. Musicians also began experimenting with the use of flat plectra (picks) on their banjos, but these were exceptions and very rare at the time.
By the 1910s it was clear more volume could be attained by using metal strings and playing with a plectrum (pick). However, banjos (in general) were still played mostly as a melodic instrument (in the classic tradition), in which a single note melody might be enhanced with chords (rather than the chord melody styles that came later). By the end of the 1910s all this began to change.
In 1916 there was a huge ukulele craze which popularised a "stumming" style of chord accompaniment. Banjoists in dance bands started to copy this new "strumming" style.
The banjo moved from being a melody instrument in the Ragtime era to a rhythm instrument in the Jazz Age. To pick out or strum chords rapidly on a gut string banjo with bare fingers was not ideal - especially at the tempo most of the 1920s music was played. The banjoists would need to be highly accomplished to do so and even then would risk being drowned out by the volume of the other instruments. Banjos needed to play much louder to be heard effectively in dance bands and so not only did metal strings became essential, but soon resonators were added to give even more volume. Besides playing finger style (without picks) on metal strings at the speed the music went then, would have meant risking serious injury. So using a flat pick became standard.
Banjoists who had already been playing the fingerstyle tended to prefer to leave off the 5th string as it only gets in the way when strumming chords, especially when playing in the keys most dance music was played in. That is, in keys which suit trumpets, trombones and saxophones. So as the musical style changed, the technology in instrument building changed.
And new banjoists tended to learn the tenor banjo. With a first string tuned higher than on 5 string or plectrum banjos, it could "cut through" a large dance band much easier. Also, with its open 5 tuning it was relatively simple for violinists, or mandolinists to learn.
Most banjos were, by the 1920s, being built to be strung with metal strings and have resonators, as that was what was most often required of them. 5 String and open back banjos didn't disappear, but compared at the time with their four string counterparts, were now seen as a bit old fashioned.
Meanwhile, the same craze for Hawaiian music that swept the world in 1916 and caused the ukulele craze, also raised the popular awareness of the Hawaiian guitar. The original acoustic instrument that is, not the later pedal steel guitar which evolved from it. This was about the only instrument at the time to use individual finger picks. These, together with the slides and bottle neck began to soon be incorporated into blues music - particularly on guitar. The same experiments in the 1880s and 90s with steel strings on banjos had also given birth to steel strung guitars that today we would refer to as Parlor guitars. Hawaiian guitars had evolved in part from these technological developments. Blues guitarists would take elements from both the classical and Hawaiian styles and create their own style. Again another story.
Anyway, the idea of using the finger picks used by Hawaiian guitarists, on metal strung banjos began (in the mid 1920s) to catch on with some (but by no means all) of the big name Classic Banjo soloists in Vaudeville. Individual finger and thumb picks would eventually become part of the "new" Bluegrass style which came to prominence in the 1940s.
So the reason a flat pick is usually used with a plectrum banjo is because of the context in which the instrument was developed, and the music that is associated with it. Finger picks as a general rule weren't used with banjos at the time the Plectrum Banjo was first made. And it was essentially made with the objective of being a strummed instrument, played with a flat pick. Of course it is not and need not be limited to strumming only, not for that matter flat picking.