Sensible shifting on a triple chainring?
Less complexity (less things to go wrong and set-up.)
No chance of shifting front and back gears simultaneously leading to
loss of chain tension and resulting in chain drop.
Less weight.
Optional chain guides.
More clearance (in most practical applications)


Furthermore:

3x7 is very old. 1x11 is pretty new. Materials and technologies have evolved significantly since 3x7. Naturally this is not an inherent advantage in the idea of 1x11 but an advantage of real world 1x11 over real world 3x7.

However:

1x11 has worse chain line than a correctly used 2x? or 3x?.

You should generally avoid doing simultaneous gear shifts. Most people who think they are often doing simultaneous shifts really shift the rear, then shift the front a very short time afterwards (or vice versa, though rear then front is far safer). 

Performing genuinely simultaneous shifts, especially on real MTB terrain, is asking for serious damage to yourself and your bike.

Finally:

The actual choice is largely personal but is influenced by terrain and riding style. 

I for example love 1x11 but it is totally impractical for my local area + style:

15degree-20degree climbs and some even steeper descents. Some descents don't really give you much opportunity to pedal, but others do and the 1x11 can't provide a high enough gear. Likewise some climbs are not particularly technical but very steep and I prefer to leisurely spin up them at 110rpm which is simply not going to happen with 1x11 without making the higher gears even more unacceptably low.

EDIT UPDATE: Based on personal experience/partly subjective since going 1x11, you don't ride where I do, how I do, or in the weather I do, so your results will almost certainly vary:


With a clutch RD and Narrow Wide chain ring: No chain drop in 3.5k+
km. 
Less to do when shifting. Not a big deal, but it's nice. 
Better positioning for my dropper trigger. 
Can't maintain just the right
cadence so easily. This is worth it for me, but can be irritating if
I am riding something not very technical. However I've taken to
forcing myself to push the harder gear (initially I would drop to the
easier gear and border on spinning out), so one can also "look at the
positive side" of this feature... but it's certainly a negative.
The shifting does actually feel allround better than my old XT 3x10 (with
SLX shifters) 
Surprisingly I am getting more life out of my first
chain. (Note: I am rarely in the easiest 2 gears, where the chain
line is pretty bad.) 
I am getting noticeably more wear on my hardest
3 gears, but everything is still running perfectly. 
More noise in the
easiest 2 gears.  
Less noise everywhere else. (Even without considering the lack of chain slap due to clutch RD)


END EDIT.