Does this machine love to vibrate and shake!

What would the valves have to do with the engine shaking?

I don't remembering it shaking this much (or maybe I was more patient with it, don't know)
Valves need checking around this mileage from what I read; I'm just thinking out loud here

No. Something is wrong. Previous models were subject to intake boots coming loose and partially separating creating un-even combustion between cylinders. I would take it to the dealership before the warranty expires.

If it is out of warranty, check the intake boots and the other historical DL1000 problems associated with un-even combustion.

Oh, yea. What mods have you made to the bike?

One thing that sometimes helps.....sometimes.....loosen the engine mounting bolts and then re-torque them properly. Loosen ALL of them to allow the engine to move a bit and then torque them back to specs. It doesn't usually help to jut try to tighten them, they need to all be loose enough at the same time to allow everything to "settle" into place before they get torqued.

Nope, my 2014 is quite smooth off idle. Remarkably smooth, the Gen 1 DL1000's I rode were like that but later (gen 2) models are very smooth and Gen 2+ bikes are supposed to be even smoother. There's a pulse at lower RPM's but it's less annoying than the typical i4 buzz. I'll admit it improved a lot with a valve job but it was nothing like what you describe even before that.

Check the valves, have a set of plugs standing by. And make sure the front plug well isn't full of water.

If there's something seriously wrong with the combustion, it could also be noticeable in the fuel mileage. Worst case scenario (no ignition in one of the cylinders) your fuel consumption could double. Any signs of that?

All those 1000cc suzuki engines had an issue with the clutch basket wearing and starting to vibrate around 3000/4000 rpm. I had a gen1 DL1000 and it was really annoying me until i got the clutch basket out and had it modified to reduce excessive play in it. My understanding is that the clutch basket design has never really been revised by Suzuki over all these years, so the 1050 might well be affected too. It takes a bit of time to manifest but after it starts wearing it can only get worse. You can see the issue I am talking about in the video below. I am not saying it is the cause of your problem, but surely something to investigate once you have tried more simple fixes like checking the engine bolts etc.

I might have made it sound worse than it actually is. The symptoms aren't severe enough to be what some of you pointed out. I don't have abnormal vibrations at idle. The engine feels underpowered in low (2-4k) rpms for such a torquey configuration.
The sparkplugs are clean and brand new.

I'll try loosening and retorqueing the engine mounts. Is there anything special to it?

@richlandrick No modifications (beyond electrical farkels, like LED, Scottoiler, etc). Everything is stock (mappings, airbox, exhaust, etc). I'll check the intakes.

@Gilles I'm aware of the clutch chudder these bikes suffer from. I might have a slight hint of that, but it's still nowhere near causing this sort of problem (and I don't the characteristic symptoms)

If the bike doesn't backfire into the intakes, it won't blow off the intake boots. Cause n effect. Chudder is a known issue that grows with age. Engine mount mis-adjustment has been found to be cause. There is a process listed in manual to address this. Detonation, or near detonation mixture leanness for sure can contribute. Uneven cylinder charging from throttle plate imbalance. Check vacuum ports. Valve clearances should be to spec. (When you pulled plugs, were they simil in color?) Pure speculation here but Suzuki called out sodium filled exhaust valves 2023. Why this? Lean burn runs hot.
We're all guessing here, but loosing fasteners is odd in its own way. Hope your patient gets well soon.

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Compared to any DL650 gen1-3 that I've ridden, my 2020 DL1050xt shakes more down near 3000 RPM. The gen1 650 had very little pull down below 3500 RPM, so you always kept revs higher and it seemed smoother. I lug the 1050 lower and it always feels very relaxed, like the engine never works to haul my ass around. So I'm very often in riding below 4000 RPM, and sometimes drop below 3000 RPM. When below 3000 RPM you feel the big cylinders bucking a bit. More shake than I ever had on a 650. Mirrors get blurry. Compared to a Harley it's very mild, but relative to the 650 it significantly more. I hope there's nothing wrong with my bike...

Forgot to mention that in 14000 miles I've rattled off the plastic left boot protector, my right front SW-Motech PRO side rack mount bolt, two rear spoke nipples (gone) and a half dozen rear spokes loose. I've spent hours at highways speeds, and many more on dirt roads.