Why can't DSLRs do HDR as well as phones can?

UPDATE:  I rethought my answer and deleted and replaced my original answer with the following (there was one thumbs up for the old answer):

Pentax does in camera HDR, and I'll put those results up against any phone any day of the week and twice on Sunday.  The images from phones look good because you're looking at them on a small LCD screen.  This is far too small for the human eye to see any defects due to poor lens design or the horrendous amount of noise that phones produce even at base ISO.  You don't know what you are giving up for the convenience of a phone.  I think if more people did, there would be a lot more people using actual dedicated cameras instead of phones.

Your question is obviously based upon your personal experience of the major DSLR brands, namely Canon and Nikon, which is understandable.  However, Pentax DSLR bodies do a lot of things that Canon & Nikon bodies don't such as in-body image stabilization (IBIS), astrotracer, and pixel shift are all things in most Pentax bodies (even the entry level ones) but not in any Nikon or Canon DSLR.  Why?  Because Canon and Nikon know that don't have to.  They know consumers will buy their cameras because of the name.  Pentax, on the other hand, doesn't have that name recognition and therefore, they have to put more tech in their cameras to entice people to even consider them over Canon or Nikon.  Have you ever considered Pentax?  Probably not because of the fantastic marketing of Canon & Nikon that promotes the idea that if you want great pictures, then you have to be using a Canon or Nikon, which is simply BS; just ask any Holga owner.  I know, I bought into the marketing BS, too.  I used to be a Canon and Holga shooter and had you told me that one day I would be shooting Pentax and loving it, I'd think you're crazy.

So, you see, DSLRs do, do the things that phones do and a lot better, too.

DSLRs don't have in-built filters.

You can also pose the same question about doing panoramas in phone or using a DSLR and software on, god forbid, a desktop computer. I recently saw a pano taken by an Apple cell phone. At first glance, it looks pretty good, but when blown up (equivalent to a 16x20" print) you can see the poor stitching job. Part of that was user error (most likely holding the camera at arms length while panning), but it's also processing issues since they are taking short cuts with the stitching to reduce processing load. The software I use to stitch would have worked much better. However, you probably wouldn't have noticed the issues since you probably don't have a critical eye for problems associated with panos.

First off, phones make anything look good on the LCD, even if the shot is poor. This is because the phone's small LCD tends to hide gross errors. Second, pictures taken with a smartphone are tuned to make it look good on-screen. Yes, computer logic helps a lot on that and it may be not as aggressive as with cameras. We'll go back to that later. Third, HDR in phones, even if you think is instantaneous, is still a multiple shot picture. You can't go away from taking at least 3 shots. You just don't feel it.

So why not dedicated cameras? First, the camera is just a tool and being so is just supposed to be manipulated by the photographer to get the picture. Skill comes in first so what you take tells a lot about you. Second, cameras are not well-blessed with the processing power of smartphones. Cameras are dumb. Again, it's the user who's supposed to be smart, right? Third, although HDR in some cameras can be made automatically doing multiple shots too like a phone, it is best done manually, because you're supposed to know what you're doing in the first place. If you don't know, don't blame the camera for getting it wrong.

There is nothing to catch up. The disparity of preference between a smartphone and a dSLR has never been wider. To each his own. Pros prefer it to stay this way until kingdom come. 

Phones have exactly the same problems as any other camera - movement leads to ghosting and they don't work well in low light (much worse than DSLRs / mirrorless).  In their favour, they can shoot at high frame rates, and the images tend to be displayed on very small screens, so the problems are less obvious.

"When will DSLRs catch up to the intelligence of phones???"

Years ago!  Much of a camera's smarts is aimed at enhancing creativity while phones use theirs to try and overcome the inherent limitations caused by tiny lenses, noisy sensors etc.

"How does the phone HDR feel so instantaneous?"

It can take the required 2 or 3 frames very quickly and there are no mechanical parts to give it any "feel".

<edit>

"When I said it feels instantaneous, I meant in the context of time, not of haptic feedback"

My point is in the absence of feedback, haptic or otherwise, how do you judge the instantaneity of something?