Controlled burns are harder to do as climate change gets worse, but that just means creating larger firebreaks around burn zones and dedicating more resources in a shorter timespan (due to the shrinking season for safe burns).
Many insurance companies in fire prone areas already require homeowners to clear their yards of fuel so with well funded, proactive fire departments, the vast majority of populated areas can be protected with a once a decade (or less frequent) burns.
Yes, controlled burns are useful at any stage. It's not like current climate is the hottest ever for this planet. In fact, it's one of the more cold ones: glaciation still exists!
by the point we leave the ice age, there's no guarantee that there'll be anything left to burn - the whole place could undergo desertification well before the ice caps melt.
It is extremely unlikely. Go to google maps, somewhere near perm and zoom to a level where you can see individual buildings. Then try scrolling east to Vladivostok, and notice how long it will take and how many trees are there.
Worst case predictions for complete loss of the ice sheets is 500 years [1] while the desertification of Mesopotamia only took a few hundred years of pre-Common era agriculture. Russia's forests are vast but we're on an industrial speed run and I've seen how fast the populations can collapse here in California thanks to some beetles and a single extended drought. Every time I visit my hometown I see the same thing happening in Russia in an ecosystem and society that are, like California and its chaparrals, completely unprepared for increasingly hostile weather.
The example I had in mind was the Sahara, or even just the evidence that the U.S. modwest has gone through significant periods of time as a desert. In ye olden days there might have been rainforest at the poles, but near the equator things were less lush.
Many insurance companies in fire prone areas already require homeowners to clear their yards of fuel so with well funded, proactive fire departments, the vast majority of populated areas can be protected with a once a decade (or less frequent) burns.