Using a sprayer isn't going the eliminate the problem of the obstacles. If anything, it makes it harder. You need to tape and dropcloth off anything you don't want overspray to cover, even with a HVLP. HVLP doesn't eliminate overspray, it just minimizes it. Using a brush or roller will let you target your need to be painted spots a lot more accurately. WHere a sprayer will work great is for large same color expanses like walls, but even there, you will still have to tape off the trim. You can use a spray shield instead of taping, but you need enough experience to be able to maneuver that with one hand and the gun with another. It's not for a novice. It takes practice to get right and starting with a hard job isn't the way to gain confidence and control.

From your description of the job, I don't think that a sprayer will help you make this an easier job in any significant way. 90% of painting is always the prep, and only 10% is the actual application of paint. Speeding up that 10% by adding another hour or two to prep doesn't leave you with any gain. All it does is leave you with a less quality job (sorry) if you've got an inexperienced person operating the sprayer for a difficult job.

If you think you have enough paint jobs in your future to benefit from a sprayer purchase, then purchase one and start on easier jobs first to gain experience. However, most homeowners that I know who've purchased their own sprayers use them for a single job and then never touch them again. The "hassle factor" is high with a sprayer and it's just easier to use a brush and roller, even if it's less "cool".

We already figured on taping plastic over the walls. But we need to paint the ceiling, which is exposed joist bays with ductwork, wiring, etc. We want to blast it all with one color. Do you really think a brush/roller is going to be easier to do that than a sprayer? I have to think a sprayer is going to be easiest, no?

Different types of paint sprayers are used for different jobs. If you will be applying house paint you will need a spraygun for that application. If you are spraying a piece of furniture with a clear finish, you will need a different gun or needle for that. My suggestion would be to tape off your room, cover everything with plastic, taped down, then go rent the sprayer. (read and understand the directions first. No need to pay for reading how to use the equipment or for taping off the job.) That will give you and idea of if you want to spend the money for one.
I own Fuji HVLP sprayer.

I just bought a Wagner Control Spray paint sprayer from Ace for $80. I looked at HD and the better ones (Graco) started at $200 up to $1000. They also had systems that involved a separate air compressor and paint attachments. I had never used one before. I figured if I liked the Wagner and used it enough I could upgrade later.
I tested it on cardboard and was surprised at how little over spray there was and loved the even coat. The fluid container is small, so not sure how practical it will be for bigger projects. Clean up was easy and it was lightweight and quiet.

I've never used a sprayer but I hired a guy once to spray the gutters and aparently he didn't know how to use one either. I showed what he did to another painter and he said that's what happens when you aren't experienced with a paint sprayer. It's hard to describe but there was a bumpy bubbly effect (without the air bubbles). I wish I could tell you how to avoid it but at least you know what to look out for. Perhaps he wasn't moving the wand evenly across or stayed in one spot too long, all I know is that it looked ugly and that was the last time a sprayer has been allowed at my house.
Now my neighbor's experience was much better. She had a lot of painting to do so she bought an inexpensive sprayer (like those Wagner things) and said it was so awful to use she returned it and paid the $200 or $300 it cost for a decent one. But in her case it really was money well spent for all they had to paint. Her husband was in charge of learning how to use it and he did say it took a little practice but once he got the hang of it the result was perfection. He painted a house full of doors and all the interior walls in no time flat.