Actually, American manufacturers (notably Conn and Olds) produced slide-tuning instruments well into the 1960s. The design lasted longest in bass trombones (the Olds P24 and Conn 7xH). S. E. Shires currently produces a slide-tuning alto and makes slide-tuning slides for tenors and basses on special order. Kanstul makes a slide-tuning bass in its 1662 series and 1670 series, and a slide-tuning tenor in its 1688/1608 series.
Slide tuning has a substantial advantage for bass trombones because it allows a longer continuously tapered bell section. Trombones are roughly 1/3 tapered and 2/3 cylindrical tubing. A Bb/F bass trombone with slide tuning (and NO provision for bell tuning) can have a tapered tubing proportions that are approximately correct for the F side. Of course, then it's too long for the Bb horn. Slide tuning also allows for a longer slide section. The Conn 62 has an in-tune low C without worrying about going off the inner slides. Other instruments (notably the Bach 50) have the low C (on the F side alone) perilously near the end of the slide. Shires went to slide tuning on his alto so that the instrument would have solid low A and E. Other altos are notorious for having these notes almost off the slide.
Slide-tuning has two primary disadvantages. First is weight: including the tuning mechanism in the slide necessarily increases the outer slide's weight. More weight means more inertia means a slower slide. Modern designs have much lower weight penalties than the designs of the 1960s. The other problem is slide alignment. A slide tuning instrument has more moving parts, which theoretically make it vulnerable to alignment problems.
As with all things musical, it is a matter of paying your money and taking your choice. Slide tuning and bell tuning trombones have different timbral and playing characteristics. Having a conical bell section generally gives a warmer sound than that of a bell tuning instrument. However, the conical bell section is very open and may require more air and effort from the player at louder dynamics.