You are right, it is because they are using very old horns known as natural horns which had no valves or holes. They had known limitations and were quite difficult to play. You can see and hear a natural horn here:
Here is an excerpt from a review of the show which mentions it:
However, because they were playing on historical “natural” horns (i.e. with no valves or holes), the tuning was often at odds with the string section, which came as a bit of a shock to modern ears. In the second suite, the antiphonal effect of the trumpets and horns was spectacular despite some ensemble problems – nine horns or nine trumpets playing completely in unison is difficult even on modern instruments – of course, one may argue that this was a part of the historical charm.
Here is an excerpt from the Wikipedia page on natural trumpets which gives interesting information about the decline of natural instruments:
Baroque composers—such as Antonio Vivaldi, Georg Philipp Telemann, George Frideric Handel and Johann Sebastian Bach—made frequent use of trumpets in sacred, orchestral, and even solo works. Many of these trumpet parts are technically quite difficult to play on a natural instrument, and were often written with a specific virtuoso performer in mind, such as Gottfried Reiche (Bach's chief trumpeter and the subject of a famous painting of the era) or Valentine Snow, for whom Handel composed some of his more noted trumpet parts. Indeed, highly skilled trumpeters were a prized commodity in the era, held in high esteem and avidly sought after by musical patrons.
The vast majority of baroque trumpet parts were written for a natural instrument pitched in C or D, although there were occasional exceptions. J.S. Bach, for example, calls for a trumpet in B♭ in his Cantatas Nos. 5 and 90, trumpets in E♭ in the first version of his Magnificat and, most famously, the solo trumpet in high F in his Brandenburg Concerto No. 2. In the 18th century various attempts were made to overcome the limitations in the notes available to natural trumpets. As early as the time of Bach, crooks (additional lengths of tubing) were introduced between the mouthpiece and the body which lowered the pitch of the instrument and allowed it to be used in a variety of keys. In the latter part of the 18th century side holes covered with keys and a sliding mechanism were tried. Later Anton Weidinger, court trumpeter in Vienna, invented a 5-key trumpet. These experiments were not completely successful, however, since side holes, which work well on instruments with a conical bore, such as cornets and bugles, cause a muffled sound in those with a cylindrical bore.
Natural trumpets continued to be used through the Classical era and even into the early Romantic period. But changing musical styles along with a growing dearth of sufficiently capable players spelled an end to the high, florid, complicated parts typical of baroque music. A few transitional composers, such as Michael Haydn, Leopold Mozart and Johann Molter wrote concerti for natural trumpets in the early Classical era. In fact, it could be argued that the concertos of Haydn and Molter represent the zenith of the natural trumpet in terms of technical demands, containing as they do some of the highest notes ever penned for the trumpet in symphonic works (in the case of Haydn, a G above high C - the 24th harmonic on a natural instrument).
Dr. John Ericson of Arizona State University wrote a book on the subject and made the video above. Here is an excerpt from an article he wrote on the technique involved in playing a natural horn, which can be observed in the video.
Modern brass instruments generally possess better intonation than comparable early instruments due both to their more efficient internal tapers and the avoidance, through the use of valves, of the most out of tune harmonics, which had to be used on the natural horn to obtain a complete scale.
Players discovered that by inserting the hand into the bell of the horn they could alter the pitch of the instrument for improved intonation and additional pitches, and that they could also make the tonal color darker and more mellow.
You can find the complete article here.