If you're avoiding treble, HD650 should be top on your list. That's it's fame (and it's bane) is its rolled off treble. Many people hate it because it's "not exciting enough", and many love it because it's "wonderfully refined and fatigue-free." If you like it warm & laid back, HD650 is certain to please.
Sound stage-wise, they're more similar than different really, and it probably depends more on amplification than which headphone you get. Being more flat, HD600 may come across as a bit wider, and being more warm HD650 may come across with a more coherent center image, all in your head of course. But that may be compensated as the HD600 drivers sit a bit closer to the ear. I'd say soundstage will have greater difference between amps than between these two headphones.
Also, in response to your original question about tubes (though answered already), it's purely about preference. There are many HD650/600 owners who swear by tubes who like the smooth "liquid" sound they create, and others who insist the headphone is "liquid enough" and benefits from SS sharpness to liven them up, so it's purely 50/50 preference. These headphones do respond well to the difference between the two. My favorite amp with them is the my Schiit Lyr (tube/hybrid) with a collection of nice tubes. I disliked my Headroom Micro SS (congested mids), moderately accepted the O2 (too neutral, analytical with HD650, IMO.) But on my Marantz PM6004 (big speaker amp I bought for HE-6) they actually sound quite good, similar to the Lyr tubes. They take advantage of the differences between amps very well, so it's more about finding the sound you want than "good or bad" types. It's picky about what you pair it with, but there are valid choices at every price point. It's not so much about "needs expensive amping" so much as "needs one it likes at a given price." Obviously, it scales well (both 600 & 650) to expensive amping too, of course!
