Have there been fully pneumatic aircraft?
Yes there have been, and some of them are still flying.

This article is about the fully pneumatic actuation systems on board the F27s built by Fokker and by Fairchild. If there was a leak, it was leaking clean air. System pressure was 3,300 PSI = 228 bar.
Some of the advantages cited:

Many factors favored the use of pneumatic systems on the F-27, most significant of which was an approximate 100-lb weight saving over electrical and hydraulic systems.

Lighter weight
Faster operation
No flammable liquids
Cleaner and easier to maintain


The systems were designed 50 years ago, fully aeronautical lightweight design. Could most definitely be used to pump up the tyres in this question to 16 bar.