I can't see an e-ink Kindle working that well for magazines, unless the text is reflowable and you don't care too much about the pictures (or you splash out on a colour one). And given you want something handbaggable, if the magazines are stuck as non-reflowable pdf-style pages, reading them on anything smaller than a large tablet is going to involve a lot of tedious zooming and panning.
If the magazines do reflow the text on screen (and you don't care too much about the pictures) then sure, a Kindle would work if the magazines you want are available on that platform, but I second the concerns others have about the locked-down nature of Kindles.
I don't read magazines on my e-reader, but wanted something I could easily pocket or put in a bag, and use to read books from a variety of sources, so I use a small phone-shaped Android-based e-reader (a Moaan Inkpalm 5 Plus; was about £85 from AliExpress during a sale with some coupons). I've installed apps like Moon+ Reader and KOReader for my own ebook files, Libby and Borrowbox for library ebooks, and the Kindle and Kobo apps to read books I've obtained in the past through those services, as well as a couple of useful tools like F-Droid for downloading open-source apps, and a better browser (not that it's a fun browsing experience, but I might occasionally download books from websites). I think one or two of the Android reading apps are optimised more for phone screens than for e-ink, but you can often improve the experience by doing things like turning off page-turn animations. It's much more versatile than the Kobo I had before (it's basically an extra-light, e-ink, wifi-only, speakerless phone), but without the temptation to doomscroll or play mindless games that there is with a phone. And in most of the reading apps, I can tell them to use the volume buttons as page turn buttons, which my old touchscreen Kobo didn't have.
Edit: this is a bit me-specific, but another reason I personally don't like Kindles is because the text formatting is too restrictive. I'm extremely short-sighted and like to read in bed, without glasses digging into my face. I have to get text within 10cm of my eye to be able to read it, which means, firstly, normal-size text looks huge, and secondly, to read normal-length lines of text, I have to constantly swing my eye massively from side to side, which gets a bit wearing.
E-readers are great for my situation partly because I can get them close to my face in bed without having to balance half a book like a tent on my temple, rest my face on half a book, or strain to hold the book fully open/fold the book back on itself (and occasionally twang myself in the face). But the other benefit is that I can adjust the text size so it's ludicrously tiny, and adjust the margin size so they're ludicrously wide. I could easily do that on a Kobo, and my Inkpalm is already very narrow and lets me set text size to whatever the reading app I'm using allows.
But any time I've fiddled around with a Kindle to see if I could make the text comfortably readable without my glasses, it was clear that Amazon had just arbitrarily decided that I shouldn't be allowed to do that.