TL;DR - Pretty much everything in your post indicates that you don't really need a new camera, but need to learn how to use the equipment that you have.

I haven't used RAW... it still boggles my mind a bit that the RAW to JPEG conversion can't have an option to do this automatically.

This is like driving a race car and never shifting past second. The amount of information available in a RAW file is astronomically more than a jpeg. The conversion to jpeg kills off some data - exactly what data you care to keep is an artistic question and responsibility of the shooter - not something to be outsourced to a camera software developer.

I can often just "point and shoot" and get a great picture, whereas even with a lot of fiddling with manual settings on the DSLR - assuming that the moment hasn't passed - I often get photographs that look inferior.

Your phone is assuming that you want a certain style of photo, and it's apparently not wrong (for you, in this case). A DSLR makes no such assumption - you need to tell it what you want. How does it know whether you want a blown sky or not?

Additionally, your phone is doing some quick edits assuming that your end goal is viewing on a phone...a tiny phone. While this is largely true for most phone photos...many DSLR shots are destined for larger viewing, possibly even printed and hung on the wall in sheer 24"x36" glory or larger. Lower quality can often be hidden with images viewed at small sizes.

If you want to do some quick and dirty HDR later, you don't really need the tripod even. Turn on Auto-Bracketing and Multishot and simply hold the release down for 3 frames. You may move a tiny bit that won't align in all photos, but you can just crop this little bit off.

You don't have to set up a tripod, take careful meter readings, align a gradual neutral density, etc. These are options available to you if you so choose - so is the quick and dirty method...and there are pros and cons to all of them.

Exactly how you choose to combine the HDR is also an artistic choice. It's on you to learn how to use the tools of your choice to get the result that you want.

I like full control to be there if I need it (and even being able to program shooting sequences in some kind of scripting language if that was possible) but I find that I often do this just to try to get around issues I have with the camera. For instance, it's not uncommon for me to overexpose by 2 or 3 stops when shooting objects in the sky like planes or birds so they don't turn out underexposed.

You're full of contradictions. Shooting RAW is full control, yet you don't? Shooting in Manual mode does open up control, but it is wasted if you don't know how to use it. I guarantee you that shooting a bird in the sky is possible without blowing the sky by 3 stops, while still getting great detail on the bird, and without using HDR.

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I don't think you need to go shopping just yet. Your post indicates that you don't know how to get the most out of your current gear and are not comfortable in post processing. Solving these two issues will yield drastically different, and better, results than simply throwing money into a new rig.

Start by learning about metering, exposure, and how to read a scene. Learn how to use a histogram. Definitely shoot RAW.

If there is some result you are trying to get to but you can't figure it out in post, then ask on this site or others how to get your file to where you want it to be. Many people out there will start with your RAW, edit to what you're asking, and then post the steps to the process. The more you do, the more you learn. It all becomes second nature at some point.

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Edit to add:

I did a panoramic shoot yesterday with over 2000 pictures... no way I have enough SD cards to store all that RAW, or time to process NEF to JPG on that many pictures.

Yea, that happens. These are the times when I would shoot a grey card, custom set white balance, exposure, vibrance/contrast/sharpness settings...shoot a test and check the histogram and really dial it in. By all means, go for jpeg in these cases - just realize the trade off that you are making.

As for processing times...many a shooter has figured out some defaults that they like and applied them in batch and let the computer run overnight to process. One of my first jobs was all of the editing for a studio - 8 shooters running portraits all day with weddings and families on the weekends. Trust me, 2k may seem like a lot of images...but learning your software better will help - they all have automation features.