Do wingtip vortices cause a decrease in the lift component of aerodynamic force, and do planes need to pull more AoA to compensate?
What you know about wingtip vortices is what can be found all over the Internet, but it is not really true. It is like saying that wet streets cause rain.

Your way of explaining things confuses cause and effect. The vortex is the effect but not the cause of drag-creating flow. And lift-creating flow, by the way. But nobody claims that the tip vortex is the source of lift, which is equally false and uses the exact same logic.

Wingtip vortices are just the tip of a full vortex sheet which leaves the wing. This vortex sheet is the consequence of the wing accelerating air downwards. Now instead of repeating myself all over again, please allow me to point you to the many other answers which explain what is going on:


This answer explains in detail how lift is created and how the downward acceleration of air is accomplished.
This answer goes into the formation of the wake and the wingtip vortices.
This answer shows why more wingspan reduces induced drag. It matters little whether the wingtips are turned up, down or continue straight: A larger wingspan reduces induced drag.


You will find that your picture of a tilted-back lift vector is correct, and indeed will a narrower wing experience larger tilting. Also, to achieve the same lift, a narrower wing needs more angle of attack and will create more induced drag. You are right again here, but instead of an ever increasing snowball effect, drag is just growing with the square of lift when the angle of attack is increased.

Ideally, all aircraft would have infinite wing span. But that would drive wing weight up a lot, so the designers settle for some compromise where the wing creates the least drag for a given net lift (total lift minus the lift needed to carry the wing).



Now to the questions in the comments:


  so the point of a higher AR ratio wing isn't to reduce vortices... It's simply to reduce the required AoA that is needed?


The point of a higher AR (more precise: More span per lift) is to involve more air in the creation of lift. This reduces the strength of the bound vortex, so indirectly it is about vortex reduction. The AoA reduction is not an issue here - in inviscid 2D flow an airfoil produces no drag, regardless of AoA. What does produce drag is the vortex flow field when span is limited.


  the reduction in vortices is simply how we see that it is working, per se?


Yes, reduced bound vortex strength results in reduced wake vortex strength.


  why do [designers] also care about the chord?


For that, allow me to refer you to this answer, and maybe this, too.


  Is it correct to say that we can measure the energy an aircraft loses due to induced drag by measuring the energy in the vortices?


Yes. But measuring the vortex strength is harder than measuring the downwash angle over span, which gives the same result. For viscous losses this is indeed a practical way of measuring drag by use of a wake rake (see the answers to this question for more).