Yup, DSC and TOS story and lore fit together just like Batman '66 fits together with Batman Begins.
You go to the doctor for a heart transplant, and they give you a kidney. They explain, "I told you I'd give you an organ from another human and this is an organ from another human. It's not like I put a balloon animal in your gut instead. Kidney and heart are quite similar. They're both human organs. In fact, from now on I'll be calling the kidney a heart, let's just call all human organs hearts. You'll be fine."
It's Trek, none of our lives depend on it, so the fun is in our enjoyment and our discussions. I like Batman '66 and Batman Begins pretty similarly. And yes, the lore can be fitted together with a minimum of mental gymnastics. There doesn't need to be consistency with prior Trek, so there's no need for trying to pommel horse DSC to functionally or tonally fit with TOS.