How was the potassium battery invented?
The search for a suitable replacement for lithium ion batteries has been ongoing pretty much since the invention of lithium ion batteries. However, the motivation for this research has primarily been cost, and shorter charge times (along with other advantages) is simply a fortuitous property of the cheaper potassium ion batteries.  

Part of the reason that potassium ion batteries are much cheaper that their lithium counterparts is the relative prevalence of potassium in the Earth's crust, potassium being present at levels about 900 times that of lithium.  

According to this Wikipedia article on the development of the potassium ion battery:  


  The prototype device used a potassium compound Prussian blue as the cathode material for its high electrochemical stability. The prototype was successfully used for more than 500 cycles. A recent review showed currently that several pragmatic materials have been successfully used as the anode and cathode for the new generations of potassium-ion batteries.
  [...]
  The potassium battery has certain advantages over similar lithium batteries (e.g., lithium-ion batteries): the cell design is simple and both the material and the fabrication procedure are cheaper. The key advantage is the abundance and low cost of potassium in comparison with lithium, which makes potassium batteries a promising candidate for large scale batteries such as household energy storage and electric vehicles. Another advantage of potassium-ion battery over lithium-ion battery is the possibility for charging faster. This means that the next generation of mobile phones based on potassium-ion batteries can be charged within a few minutes only.  


So, while faster charging times are certainly a desirable characteristic, this was not the driving force for the development of potassium ion batteries.