Type 6BS7 is essentially a type 6BR7 with the signal grid taken out to a top cap in order to overcome the external 'hum problem'. However, hum was not a problem to professional hi-fi designers who preferred the single-ended 6BR7 for other good reasons and (on the whole) ignored the 6BS7.
Moreover, ham-fisted DIY enthusiasts who could not handle the 6BR7 often could not the 6BS7 either (since hum is not the only problem in high-gain audio amplifiers). Above all, it makes no sense to design and market a valve specifically for incompetent customers! As a result the 6BS7 was much less widely used.
R J Sutherland has been looking at an old Brimar data book and found they recommend the 6BS7 for use in valve voltmeters, pH meters and other electrometer applications. In an example electrometer circuit with an input current of less than 5 pA. Such high input impedance would require very careful circuit design and layout and would pick up any extraneous signal. The 7851 is similar but was designed only for use as an electrometer valve. The 7851 has a 2.5 Volt heater and is designed for very low anode voltages.
Looking into the plane of the grids.
The working face of the anode.
The thin glass tube envelope is 21 mm in diameter and, excluding the B9A base pins, is 50 mm tall.