The artillery was on a schedule. The barrage had to move in such a way that it advanced in front of the infantry suppressing the defenders. To achieve this the British counted on 10% of their attacking force being lost to friendly artillery fire. The artillery had to fire move fire move and could not change this to worry about a counter barrage.

At the battle of the Somme 1.73 million rounds were fired by 1400 guns. That's 1250 rounds per gun in 8 days. The rate of fire was between 2 and 4 rounds per minute. So each gun spent 400 minutes during that 8 days actually firing.

Remember this is a rolling or creeping barrage so you aren't just firing. while you are firing the gun next to you is changing it's trajectory to move forward 100 meters the one next to that is moving forward 200 meters and so on. As the battle progresses you may have to move the gun forward which takes several minutes. So yes when you were firing you just fired. Because your gun was covering an area of the line for the guns around it as they prepared to do the same.

Given the context it's probable that these artillery were firing in support of an offensive and in that case it makes perfect sense that they'd fire as fast as possible. In their first offensive after the stalemate had set in, at Neuve Chapelle, the British used 'hurricane bombardment' tactics--putting as many shells as possible into a very concentrated area in a short space of time, just 30 minutes or so. This was done because the British didn't have enough shells in the early stages of the war to fire for a longer period; but the result was to hit the defenders with fire of such intensity that they were totally neutralised, having had their defences so badly damaged and taken such a psychological shock that they couldn't put up good resistance. Additionally there wasn't time to bring up reserves before the attack came in, so the German front line was taken with relative ease. The British attempted to replicate this success at Aubers Ridge in May 1915 but the German defences had improved without a comparable improvement in British artillery, which still relied on guns which were too light for the job. The failure at Aubers Ridge led the British to adopt long bombardments of days or weeks, and these tactics continued up to the end of 1917. Although this decision has been criticised, and it's fair to say that it didn't provide a solution to the deadlock, these tactics were based on successful French methods and allowed considerably more ground to be gained in attacks; so I'd argue that they were the appropriate methods and the best that could be managed with the resources and techniques available. By late 1917 the British had achieved 'silent registration', which meant that with the use of maps, artillery boards and range tables they could aim their guns on a target with reasonable accuracy without the need to fire ranging shots first, which could alert the enemy. At this stage it became possible for artillery to become much more responsive and fire accurately on targets as they appeared with little preparation. Throughout the war several types of 'barrage' were used--this is a distinct definition meaning a screen of shells that is designed to neutralise the enemy and protect the infantry advancing behind it. These could jump between enemy defensive lines, providing cover while the infantry positioned themselves to seize the positions once the barrage lifted, or 'creep' a certain number of metres every minute, with the infantry following closely behind.

To get to the point: in all these cases, it was favourable for the artillery to be firing as fast as possible because a higher concentration of shells was always better. Gun crews firing for routine harassment of the enemy probably wouldn't be quite as concerned about maximising their rate of fire; but they'd also be less likely to be filmed and it would be less likely for this film to be used in a documentary. There's also the possibility that some of the footage is staged (like the two most commonly used clips of soldiers going 'over the top'), in which case a higher rate of fire naturally looks more impressive.

Counter-barrages weren't a major concern, or something that could effectively be dealt with if it did happen. All but the lightest artillery was essentially immobile due to the weight of the guns, the great quantity of shells that also had to be carried around and the exigencies of aiming them properly, which required batteries to be placed in known locations. Crews would construct batteries, bring up stores, dig dugouts and then remain there for as long as was possible and necessary. Dodging counter-battery fire was totally impractical, and the best that could be done was for the crews to take shelter in their dugouts if enemy fire came in. This wasn't too much of a problem, though, because counter-battery fire was so problematic to carry out. If an enemy battery was located (they were usually well concealed) this would have to be accurately plotted on maps (aerial photography could produce errors of hundreds of yards), and then ranging shots would need to be fired and observed. This gave ample time for crews to realise what was coming and take cover, and landing a shell close enough to a battery to actually destroy the guns demanded precision which artillery of the time simply didn't have. For much of the war both sides were content for their counter-battery fire simply to neutralise the enemy gunners for as long as possible, because destroying them outright was so difficult. Batteries became a favoured target for gas shells because this was more effective than high explosive at driving the gunners away or at least forcing them to work wearing their gas masks. The situation changed in favour of the British in the later stages of the war; by then their sound-ranging and flash-spotting techniques were so advanced that it was claimed that before the Battle of Amiens in August 1918 90% of the enemy guns in the sector had been identified. German artillery in the final stage of the war could rarely fire off more than a few shots before it came under counter-battery fire. However, assuming the artillerymen you saw in the documentary were British, they would have relatively little to worry about, since German techniques were never as advanced. Counter-battery fire was a problem that couldn't easily be prevented, so it was just as well that it was usually so limited in its effects.