My personal preference is for solid state amplifiers. I am not going to mention any specific amplifiers because I never recommend things I haven't had a lot of experience of myself.
However I do have a pair of K701s myself. These headphones are extremely transparent and will sound at their best with good quality transparent amplifiers.
Avoid, if you can, amplifiers that colour the sound. These have become very popular even though they are really not hi fi imho. The initial impression of coloured sound can be very good but you will soon tire of it and if you replace the coloured amplifier with a neutral amplifier it is a joy to hear your music without the added colour.
Remember that neutral transparent amplifiers allow you to hear the colour that is actually in the music!
In your post you mention "good extension, power and timing" all three of these are associated with good quality solid state amplifiers.
The next thing I'd suggest is that you consider second-hand amplifiers. For the same money as a new amplifier you can get very much better quality. It might be worth you while to look at some online auctions and get an idea of what is available.