Go do a search on Youtube for the following:
AA GK Very Difficult BST Solo
AA MR Very Difficult BST Solo
AA HM Very Difficult BST Solo
Most people would have to take such a fight very seriously, but you have this guy doing Ninja job emote since his pet is doing all the work. Now let's compare his pet's capabilities to that of players. Notice that the distance nerf did nothing.
First off tanking.... his pet doesn't get buffs other than food and gear.... so take the greatest geared PLD with only comparable food that the BST used, no phalanx, no reprisal, no protect, no shell, etc. Now then only allow the PLD to be cured as often and for as much as the BST can do for his pet. I guarantee you the PLD would die, especially when the Very Difficult AA gets below 25% HP.
Secondly damage dealing.... again, his pet doesn't get buffs other than food and gear.... so take the greatest geared SAM with only comparable food that the BST used. To be really fair in this comparison, the SAM should have to TP in damage taken gear... but if you do the math, even not being in damage taken gear the SAM would be whiffing like crazy. There is no way the SAM would come even remotely close to the damage that pet did.
So we have a pet not only greater than the best possible players in their specific functions, it is tanking and doing more damage AT THE SAME TIME. Other multi-function jobs do not get decried as much, for instance RDM, because RDM pales in comparison in the functions they are performing compared to those who specialize. But here we have a pet not only doing better than jobs that specialize in those functions, but doing multiple functions at the same time better than the specialized jobs.
This is why BST needs to be nerfed.
A good way to start on the right path to making the job more realistic is to make it so the pet actually has to have TP before it can do a TP move.