Fascinating question that I quietly wrestled with for the past 15 years. (Incidentally, I thought that @peterff made some excellent points, and I will try not to repeat what has already been said.) In order to fully appreciate my perspective, I need to give you a little of my history with photography.

I have been serious about photography for the last 40 years, but I would not call myself a serious photographer. I am not a professional; just a committed amateur.

In the prehistoric days of film, I was a Canon guy. I owned two Canon SLRs in my first 20 years. They were magnificent machines producing crisp, beautiful images of everything I photographed. I experimented with film, slides, black-and-white and infrared. In the thousands of images I took on film, there was one recurring theme: by and large, my photographs were terrible! I was young and poor, and the worst part of it all was the cost of film and developing.

By the time digital emerged, I was out of school, working and had some disposable income. I was fascinated with the digital medium because of the instant feedback and unlimited “free“ images. However, the ultra-low resolution and lack of features suggested that digital was nothing more than a toy and would likely disappear. (Whew! Was I wrong about that!)

In 1998, when Sony crossed the resolution line into the 0.3 MP (300 K) territory, I jumped in. With that, my 500–600 images annually grew to 2,000-3,000. What was most significant about my move to digital, and what kept me going with 10 digital cameras over the next 20 years, was the consistent improvement in my images. Digital gave me the immediate feedback to learn about lighting, composition and (old school) special effects. But back to your question.

Now a veteran of numerous point-and-shoot cameras (Sony, Fuji, Canon and Nikon) I most recently committed to a Sony alpha a7II, which is full-frame and mirrorless. (I shunned the earliest DSLRs because of a technology quirk which caused a momentary blackout when the shutter was triggered.)

Why mirrorless in my case? The answer is simple: weight and convenience. I get all the benefits of a DSLR without having to commit to the weight and bulk of that equipment. As one of the posters below indicates: you will never see a Hell’s Angel on a minibike. However, a Hell’s Angel is identified by his motorcycle; I am not identified by my photographic equipment. (You also won’t see a Hell’s Angel taking his motorcycle into a restaurant, or carrying it around Disney World.)

My goal is to take great images, but the world’s immediate perception of me as a serious photographer has nothing to do with the quality of my images. Just look at all the terrible film images I took for 20 years!

My conclusion, therefore, is that you should serve your objectives. If you want great quality images, just about any camera today (even point-and-shoots) can give you that result. But the subject/composition/balance is up to you. If you want to be a Hell’s Angel photographer, grab yourself the biggest, heaviest and loudest equipment you can find.

Besides me?

I had a couple Nikon SLRs that went away when film (mostly) died.

I did not embrace digital but shot family events and some travel snapshots with a Canon Elph and Nikon Coolpix.

When I decided to up the quality, I got an RX100 because the sensor was so much bigger.

After that, I got three different Panasonic M4/3 cameras.

I have no idea what I'm missing by not having a "real" DSLR!

What reasons are there that I should get one.

That decision is yours. If you have shot film SLR's and were comfortable using them then the switch to DSLR'S would not be ba big step.
The only reason you should consider one is if that style of digital camera is right for you.
Mirrorless, DSLR's, 4/3's, point and shoots, cropped senor, full frame sensor, bridge, all have their strengths and weaknesses.
It really comes down to you, and which one will deliver what you are looking for.
Personally, I use a full frame sensor for landscape and cropped sensor for wildlife, cause this is right for me.
Good luck and keep on shooting until the end.

Besides me?

I had a couple Nikon SLRs that went away when film (mostly) died.

I did not embrace digital but shot family events and some travel snapshots with a Canon Elph and Nikon Coolpix.

When I decided to up the quality, I got an RX100 because the sensor was so much bigger.

After that, I got three different Panasonic M4/3 cameras.

I have no idea what I'm missing by not having a "real" DSLR!

What reasons are there that I should get one.

Besides me?

I had a couple Nikon SLRs that went away when film (mostly) died.

I did not embrace digital but shot family events and some travel snapshots with a Canon Elph and Nikon Coolpix.

When I decided to up the quality, I got an RX100 because the sensor was so much bigger.

After that, I got three different Panasonic M4/3 cameras.

I have no idea what I'm missing by not having a "real" DSLR!

What reasons are there that I should get one.

I never owned one, either, but had all the 24/7 access I wanted to various Canon and Nikon dSLRs from 2003 until 2012, through my employers. So I can testify without a lot of bias, I think.

Generally, dSLRs are great for very high end sports and wildlife work, where high framing rates, full frame, and minimal latency (“click to clunk” time) are concerns.

Pros tend to use Canons and Nikons for the services they get from CPS and NPS (professional service organizations you subscribe to for a fee).

Pros also use dSLRs for the very long and very fast telephoto lenses that are not available on other cameras.

For years, the plentiful bounty of great dSLR lenses was almost a monopoly advantage of Canons and Nikons. That is much less a factor now, except at the very fast, very long ends of the range.

I considered dSLRs in 2012-13, but noticed what other manufacturers were doing with mirrorless. I bought Panasonic LUMIX Micro 4/3 gear for its balanced video and still capabilities, and I’m quite satisfied with it. If I ever need a dSLR, I’ll rent one, unless I need it for a whole new genre...

Bill, you should know better. It’s not possible to be a serious photographer and not shoot with a dslr !!
4:3’s are like minibikes compared to real motorcycles! Ever seen a Hells Angle on a minibike? Nor will you EVER!!! LoL
Photography is about macho factor. Even if your arms are wimpy, they just look bigger holding a pro-DSLR!
When I’m carrying my 1Dx with 200mm f1.8 attached, every other person asks me if I’m a pro!!! LoL
When carrying a 4:3 only another 4:3 guy will say anything.
I’ve seen guys carrying 4:3 get stepped on by a guy hurrying over to check out my rig.
Be a real MAN, get a DSLR!!!! Bill, are you getting my drift?!?!?!
SS

That decision is yours. If you have shot film SLR's and were comfortable using them then the switch to DSLR'S would not be ba big step.
The only reason you should consider one is if that style of digital camera is right for you.
Mirrorless, DSLR's, 4/3's, point and shoots, cropped senor, full frame sensor, bridge, all have their strengths and weaknesses.
It really comes down to you, and which one will deliver what you are looking for.
Personally, I use a full frame sensor for landscape and cropped sensor for wildlife, cause this is right for me.
Good luck and keep on shooting until the end.