When researching information for the Browns Peak hike that I did this weekend, I tried to find a definitive answer for which route to the Lone Pine Saddle trailhead was faster. I timed both drives, going up to the trailhead on Forest Road 143 and coming back on El Oso, taking 188 north to 87 and back down.
Times were measured between the turnoff to 143 off of hwy 87 and the trailhead. Your travel time may vary. I don't drive like a bat outta hell, nor an old grannie, so if you drive at fairly normal speeds your times should at least be similar.
So, as for the times:
Taking 143 off of 87 going up to the trailhead: 1 hr 6 minutes
Starting at the trailhead, taking El Oso to 188, north to 87, and south to the 143 turn-off: 1 hr 23 minutes
It appears that just about any 2 wheel drive car could make it to the trailhead using El Oso, and I saw a few 2WD cars at the trailhead. As for 143, I wouldn't recommend taking any vehicle without a fair amount of clearance, but 2 wheel drive cars with a decent clearance should have no problem as long as the roads are dry.
One last thing, look out for rattlesnakes. I ran across an Arizona Black Rattlesnake on Browns Trail. I probably would have stepped right on it had it not been a black snake on light-colored gravel, which made it really stand out.