A mouthpiece for one purpose?

[Finetales]

Notice the size (large diameter) of Reiche's mouthpiece. Looks like a trombone mouthpiece!

Yeah, the baroque trumpet mouthpieces I've played were almost trombone-sized. It's really impressive what real players can do on that setup.

[BGuttman]

Notice the size (large diameter) of Reiche's mouthpiece. Looks like a trombone mouthpiece!

Almost like the one on that baroque trumpet in the video.

[LeTromboniste]

Notice the size (large diameter) of Reiche's mouthpiece. Looks like a trombone mouthpiece!

Yes! The most successful natural trumpet players today seem to all play on really large mouthpieces, almost trombone-sized (and as far as I know surviving mouthpieces tend to be quite large). My understanding is these bigger mouthpieces make it easier to bend notes so that the few partials that are really off (especially the 11th and 13th) can actually be used when playing clarino. There's also of course the fact that natural trumpets are twice as long as valve trumpets, and playing principale requires accessing the lowest partials, with a big and clear tone.

Interesting side-fact relates to this: in the Accademia Filarmonica di Verona's collection, one finds three original 16th century trombones, including one by Anton Schnitzer the Elder, as well as "pretzel-shaped" trumpet also by Schnitzer made for Cesare Bendinelli, who was a virtuoso trumpeter at the court of Bavaria and wrote the earliest known trumpet method, which he donated to the Accademia together with the trumpet. The trumpet's bell has the same dimensions, and actually appears to have been made on the same mandrel, as the trombones's. And the bore of the tubing is actually a bit larger than the trombone's.

[harrisonreed]

The guy posing in the video Maximilien posted also had a very large mouthpiece.