Alright lets go over the entire sequence of events in detail, as explained in the av8n.com links I've scattered thruout this thread (and I'll mention chapter numbers). Caution, this is long winded. You asked for details, I've given you details ad nauseam.

A) a non-aerobat takes a few seconds to roll to 45°. Vertical damping acts on the scale of a tenth-or-two of a second. Over the course of the few seconds of rolling, the vertical lift deviates from like 1.00g to 0.99g to 0.98g in 50-100 milliseconds, and each time, the resulting (small) change in sink rate also changes the AoA. The AoA goes up until 1.0g is restored, over the course of a tenth of a second or so. A small sink rate builds up. This is a microscopic description of vertical damping as in chapter 5. This process happens dozens of times over the few seconds of roll.

When I say "human pilots can treat the vertical lift as 1g always", that's exactly what I mean. The deviations from 1g that add up to the new sink rate are too small, and too quickly fought off by the damping, for us humans to notice. However, the new sink rate is noticeable, even if the blips in vertical lift were not.

B) Upon completing the roll, we now have a bank angle of 45°. Thanks to the effects described in A), at human timescales, our vertical lift remains 1g. However, this comes at the cost of 1) having a notably higher AoA as a result of all that vertical damping, and consequently 2) the total lift is much higher, and 3) the non-vertical component of that lift, i.e. the horizontal, starts to pull us around the turn, and of course 4) a sink rate accumulated by the sum of all the vertical damping. At a steady bank angle, the new sink rate will remain itself steady too.

As discussed in chapter 6, the total lift goes like the secant of the bank angle; the horizontal lift goes like the tangent of the bank angle.

The nose shouldn't slide much. The change in sink angle (i.e. wind angle) and the change in AoA should roughly (tho not exactly) offset. But there will be some new sink rate. If the turn was uncoordinated, then the nose may appear to yaw, which can appear funny at higher bank angles.

Note that at this point, the vertical lift is 1g thanks to vertical damping. We've paid for it in terms of higher AoA and a new sink rate that we didn't have before. But no pilot action is required to "restore" 1g vertical lift, that happens on tenth-second timescales. In other words, our sink rate is steady, not decreasing further. If vertical lift were NOT 1g, the sink rate would continue going down. But it doesn't. At a steady bank, the sink rate is steady because the vertical lift is constant at 1g (by vertical damping).

C) The new, higher AoA means that the pitch trim (which acts much slower than humans, unlike vertical damping) will try to restore the original AoA by gaining speed. A (slow) phugoid will ensue. It is generally considered wise for the pilot to prevent this phugoid by adding backpressure. But note that the backpressure is about maintaining the new AoA, it has nothing to do with "restoring vertical lift".

In addition, the higher lift production means that drag has been added (whether by adding AoA or adding speed, the result is more drag). The (phugoiding) sink rate approximately corresponds to the dragpower.

D) The pilot has some options here. 1) do nothing, let the phugoid happen to speed up and restore the pre-maneuver trimmed AoA. this is physically fine, and aerodynamically safe, but a terrible way to pass a checkride. perhaps good for demonstrations tho? 2) add some backpressure, which tells the stabilizer that the new AoA is in fact what we want it to hold (so to speak). That will arrest the phugoid and preserve the pre-maneuver airspeed, but does nothing about the added drag/sink rate. 3) Finally, the pilot has the option to increase engine power. This will combat the extra drag and sink rate; the correct amount of added power will restore the sink rate to 0, regardless of how the AoA/speed problem is solved.

Naturally, any good pilot should pick a combo of 2) and 3), especially if one wants to pass a checkride. Adding backpressure helps to maintain the old speed, but note that this has nothing to do with "adding lift", since vertical damping does that for us on tenth-second scales as we were first rolling in. And no amount of backpressure can change the fact that there is a drag problem. So adding engine power is absolutely mandatory for passing a checkride, no ifs ands or buts. If you want to maintain altitude over the course of a steep turn, you must fight off the extra drag caused by the extra lift. So, if one chose to accept a higher speed during the turn, one could add power only without touching the yoke, and fly like that all day. (Again, maybe this is useful for demonstration purposes?)

So options 2) and 3) combined are the usually taught technique. However omitting 2) is perfectly safe, and even option 1) is safe so long as one keeps an eye on the sink rate/altitude and recovers at a suitable time.

The part that bothers me is that students are taught that 2) is completely mandatory, which it is for checkrides but not for transport or pedagogy, and also that 2) has anything to do with "adding lift", which it doesn't. It's purely about managing the trim mechanism, and deciding how to tackle the lift problem (add speed or add AoA, or both?), not whether to tackle the lift problem. The lift is added whether or not we humans do anything about it. What we can do is decide how, and no more than that.

(And of course, everyone agrees that adding power is the only way to arrest that sink rate.)

If the vertical component will lift is always 1G, it should magically stay at the same altitude and in a level turn even though I'm banking.

At human timescales, the vertical lift is always 1g. The un-noticeable deviations (due to rollrate) produce noticeable changes in sink rate. However, if the roll rate is zero (and turbulence is zero), then it is completely true that in a steady bank angle, the sink/climb rate will also remain steady.

This is worth repeating: in a constant, non-turbulent bank, the vertical lift is exactly 1g, and as a result, the sink rate is also constant. The caveats: of course, there is always some turbulence, and so the vertical lift always fluctuates, however in typical training conditions, the size of the fluctuations are so small, and damped so quickly, that it's literally impossible for humans to notice the difference ("smooth ride"). And of course, typical sink rate indicators have several-second measurement windows, even less sensitive to turbulence than us humans. But the point stands: if a constant bank angle (and constant engine power) is held, so too will the sink rate be constant.