I can't recommend the ski trail approach I did on Killington in June for the aesthetics of the local views, but on a clear day the views to the Whites and north to fellow 4Ks are superb if you ascend near the K-1 beeline ski lift. I enjoyed looking over to increasing views, with the climb, of Moosilauke, Franconia Ridge and, at the limit of view on my day, Washington, among other Whites. Above 3500' or so the views north up the spine of the Greens were very nice, too.
If you opt for a less disturbed, man-managed hike via the A.T. corridor, or, as with most, via Bucklin, I'd say the short side trip, 0.3 round trip, to the K-1 gondola area is worthwhile, as you get excellent SE and due south views. Nice looks at Ascutney, Monadnock and Okemo/Ludlow and cleaner views of Stratton and Mt. Snow.
The views to the Adirondacks from the summit are extraordinary. I've hiked Pico, which also has fine views, and a trip combining it, just short of 4K at 3960' or so, with K, from the Long Trail Inn and back, would make for a fun combination.