The ST500 sounds great with both my VAC Renaissance and Manley Shrimp preamps. This is with N802's. The ESP speakers are designed for a tube amps. In this case the VAC PA100 is better than the ST500. When I use the ST500 it is powered on all time (sleep mode) it awakes when a signal is present.
I recall the Sonatina's as being a good match for tube amps (non reactive and high stable impedance) so you may lose some of the tube magic. If you were attempting to drive a more challenging speaker, I would recommend you try a Wyred amp.
IME it takes maybe 10-15 min for my tube gear (VAC, Cary, Manley) to hit its stride. So, I wonder why you find it takes 15-30 mi longer for your system? I agree with Marakanetz that solid state will take longer to warmup, and even so may never equal or better what you already own.