One of the joys of studying old technology is that it takes you back to first principles and makes you appreciate the difficulties to old designers faced. Ken’s quotation " ….. no one knew what an airplane SHOULD look like" is very appropriate.
Nowadays we are accustomed to seeing refinement rather than revolution in design. It may be difficult now to understand why designers went down so many blind alleys to get to where we are today. Without knowing the barriers they faced, it’s hard to know what they were up against. In some cases available materials imposed a limit. Patent avoidance was another strong influence, then there was conservatism, lack of technology, knowledge, plain madness, etc., etc.
In the case of the leaf springs, who knows? A point to consider is this: you’re up in the sky, in a single engined plane with new-fangled overhead valves. If a spring breaks, the big valve drops. With a highly-stressed helical valve spring, a single defect can quickly grow to cause failure. A multi-leaf spring of that era may be less likely to fail, and can have more redundancy built in. Of course, with helical springs, an extra one could be nested in to provide assurance.
It also strikes me that the use of coil springs in the Beardmore would have made the engine taller, usually a bad thing in aircraft. (Taller because the valve stems would need to be longer, and also because the cylinder head would have to be beefier where the spring sits).
Rick is quite right about the damping effect of friction in leaf springs (presumably one reason why they persisted in use for car suspension?), but I don’t know whether valve bounce would have been a problem at these speeds.
I’ve just looked at Bill Gunston’s ‘The Devlopment of Piston Aero Engines’, and this has a very nice picture of this engine, but labels it as a 1912 Austro-Daimler. In fact the brass plate soldered to the copper jacket says ‘Beardmore’.