When I formulated the scenario, I broke it down into two phases.

If you mean "what happens first" and "what happens second" then I follow
you. Otherwise,

pulling back on the stick and increasing your
angle of attack enough to meet the increased lift requirement, with no
change in airspeed.

.... doesn't work.

That is, the airspeed won't change instantaneiously, but it will change,
because the steady state is unsustainable without more power. Maybe
that's what you mean by the second phase: (The increase in drag due to
the higher load factor would slow you...)

If that's what you meant, then you're ok.

and
you'd soon find yourself pitched down and descending, followed by a
restauration of the initial airspeed flown before

What are you doing with the stick? (same position? same forces? same
airspeed?)

It's slightly different, but you might be able to get your head around
it better, if you think of Uncle Bob parachuting into your airplane.
There you are, fat, dumb, and happy, flying straight and level. (I'm
going to make all the numbers up here out of whole cloth). Uncle Bob
parachutes right into your airplane, and amazingly lets go of the canopy
and fixes the airframe faster than you can say "337". But he's now in
the plane.

IF YOU DO NOTHING, the plane will descend, because lift hasn't changed,
but weight has. To compensate, you pull back on the stick, increasing
lift (at the expense of drag). Now with the nose pointing in the air,
you maintain altitude, but you're going slower because the extra drag,
and also partly because the thurst vector is now pointing more up, and
less in the direction of flight.

To go faster without gaining altitude, you must lower the nose (again)
=and= add power. You will need a higher airspeed than originally to
provide sufficient lift (you need to hold Uncle Bob up too!) if you
attempt to hold the same angle of attack as you originally had, but
that's not what you want. You want your orignal airspeed, AOA be
damned. So, at that original airspeed (which of course requires more
power), you'll need a higher AOA.

You can generate the same lift at a range of AOA and power. Doing this
with more power requires you to go faster. Doing it with a higher AOA
requires you to go slower. You pick the one combination that gives you
the desired (original) airspeed. It will be a higher-than-original AOA,
with higher-than-original power.

Most of this thinking would apply to the turning flight too, but it
might be easier to think of it in this context first. (btw, this is the
context of airdrops in reverse - you drop your load, you need to reduce
power and pitch down if you want to maintain altitude and airspeed).

Jose
--
"Never trust anything that can think for itself, if you can't see where
it keeps its brain." (chapter 10 of book 3 - Harry Potter).
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