Ankle "support" from boots is mainly an effect called proprioception: That flexible high-top collar won't physically stop an ankle roll, but because it's in contact with your ankle you feel it happening sooner and can often react quicker.
If you want real support you can wear a lace-up or velcro ankle brace like the ASO over the socks to add true lateral rigidity. Then you can use lighter footwear that reduces fatigue and pounding while making you much more agile...win-win.
I wouldn't use full boots unless I knew my feet would be continually wet and possibly freezing (long stretches of flooded trail, unconsolidated wet snow, eg.) or I needed to wear crampons.