Cutting ill branches: let them rot ok, or burn is a must?

I am pruning my avocados, but this would be the same for any variety.

I cut branches that make the trees touch each other because it stops the sun,
and I cut ill branches.

My trees are 20 years old and were never really pruned, nor treated.
A quick search showed me that they are real good for a lecture about avocado diseases! They get them all.
Mainly fungus, but also virus.

I have learned that rotting wood in a forest nourish the good fungus/mycorrizal grid of the soil.
And this grid fights against "bad" fungi.

But what's about doing this with the ill wood itself?
Will I spread more of the disease?
Take into account that I did not cut all what is ill, or else I cut everything...
So anyway the mold spores are there...

Another option is burning of course.
BUT
I will release carbon in the air for no purpose.

I would make biochar, but I am not ready, I do not yet have barrels for burning.

Any other way of burning let's say the outside and extinguish the fire before it burns all?
Will this be enough to stop disease from disseminating?

If you boil them in hot water that soil kill them.

To make the bio-char you could dig a trench and then put the "logs" in it then cover with a sheet of metal/brickwork.
At the end of the trench build a rocket stove the rocket stove intake the re-burn methane and other syngas and burn them.
The heat from the "rocket stove" will turn the logs into bio-char.

I am not sure but at 20yrs old these trees could just be old and dying and as such the illness might have less to do with horrible virus and more about them just needing to be replaced.

I would bury them in a compost heap or hugelkultur. Beneficial fungi and bacteria will kill the bad guys, in my opinion.

Thanks for your answers!
That's much more than 1 tree, so boiling logs and compost is technically excluded.
My 2 places where burying wood was possible is already done.
phytophthora (one sure disease very common here)
is supposed to come with water and infect through roots,
that is why I am not fond of burying like this...

At the moment, as I have a still unplanted little place near this wood (steep place!),
the idea is to partially burn them (the outside), so that heat kills what it can,
while retaining some carbon for next use, and then let it rot.

I just do not know how much is needed to reach enough temperature inside the wood...

Who kills who is just what I would like to be sure about!
I do not want to let wood out and let them rot unless there is real factual evidence about the way the "bad guy" fight for survival.
Any data about this is welcome...

Regarding phytophthora, you may want to look at this: http://www.ladybug.uconn.edu/factsheets/tp_05_phytophthora.html