It's been a long time, but... you can usually drive pretty far up, and just follow the USFS road to the plateau right below the top. State road 547 goes up to about 8800', then take USFS 453 up to about 10600'. The trail from there, on the NE side, will have softer snow, but is more through trees. The Gooseberry Springs route, as you've gathered, is more exposed and sloped; but Scott did it with young Kessler, on a less-snowy winter.
In the summer, a 4x4 could take the USFS road pretty high; then at a closure, you can walk a road to within a few hundred feet below the summit... at least you could in 1995! In 1983, we drove a front-wheel drive car through the snow to well over 9000', on the road that eventually reaches the summit plateau. Unfortunately, it's mainly wooded, so you don't see much on the way.http://maps.google.com/maps?q=Mount+Tay ... r&t=p&z=14
I don't know what restrictions the quadrathlon puts on access these days. Their page must have a link to the appropriate Cibola Forest office, where you might get the latest conditions