So in short, do I basically need a DSLR for "optimum" image quality?

You want to optimize more than just image quality -- you want an optimal balance of image quality and portability/convenience, i.e. something that provides image quality that's significantly better than your phone provides, but as small as possible. Control and ease of use are two other variables that you should consider -- do you want a camera that gives you lots of creative control, or one that does everything for you?

A mirrorless camera system might be the right choice for you. Mirrorless cameras give you the large sensor, interchangeable lenses, and creative control of a DSLR in a smaller, lighter package. For example, the Canon M10 is much smaller than even a small DSLR, but still packs a 18 megapixel APS-C sensor. With a 22mm f/2 pancake lens, the whole thing would easily fit in a large pocket. The EF-M lenses that fit the camera are smaller and lighter than the EF lenses that fit Canon's DSLRs, but you can also adapt EF lenses to the EF-M mount so there's a lot of flexibility. And at $500 or so, the M10 should fit into your price range.

Canon is by no means the only game in town, and you should look at the different options and decide what's right for you. The main thing is that a mirrorless system provides a good mix of image quality, light weight, and creative control for someone who needs a small camera that takes great photos.

PS: Invest in extra batteries.

Should I just deal with the extra weight and embrace my ascendance into the world of non-phone photography?

That's up to you. You're presumably familiar with the quality you get from your phone. A lot of phones have pretty remarkable cameras, but they are what they are -- tiny sensors with tiny lenses and not a lot of options or control. Having a more capable camera, whether it's mirrorless, DSLR, or a bridge camera, will definitely expand the range of photos you can take, but only you can say whether that's important enough to you to carry around the extra gear.