Doesn't hydraulic fluid suffer from viscosity breakdown?
I'd also worry about all the particles/contaminants that are smaller than can be filtered end up building within the fluid. The concern would be that this would tend to make up a greater, albeit small, percentage of the fluid: it's possible that it could also bond, which would then get collected up by the filter(s) (requiring a more frequent filter change interval- some filters aren't cheap!).
If one could get by with just filter changes I'd think that the big players might be pitching this practice to their big high-end machinery customers as that would be a huge savings (one could, perhaps, and maybe they already do this, have live/running filter switching like I believe already exists with fuel filters and(?) oil filters?- I could probably Google it!).