Okay, I'm fudging that a bit. Shakespeare combined romance and comedy. This may be why I have always thought that sex is inherently funny. The third Shakespearean genre was history.
The difference between literary fiction and genre fiction is that genre fiction has to fulfilled a defined expectation as well as tell a good story well.
In a Science Fiction story, there must be some scientific speculation that shapes the story's setting, plot and/or characters. In Fantasy, the change in the setting or characters can be magical, but, like Science Fiction, once the rules are set, the author has to follow them.
In a Mystery, there must be a mystery to be solved. In an Adventure, there has to be a quest to fulfill. A Thriller must keep you at the edge of your seat. A Cozy, not so much. It's all about avoiding the uncomfortable aspects of life like graphic violence and sex in favour of concentrating on the puzzle being solved.
A Cozy can't be a Thriller, but it can be Science Fiction or Fantasy as well as being a Mystery. A Horror novel might be a Mystery and a Thriller as well or Science Fiction or Fantasy. Lots of genre-mashing is possible. I know. I've mashed them. I've got a Western Historical Romance that's also a Cozy Mystery (complete with small town and limited suspects): Under A Texas Star. Then there's my Near-future Detective/PI Mystery with elements of a Cozy and Procedural: Deadly Legacy.
A Romance is not a Romance if there is no Happily Ever After for the main characters. It might be a literary/historical love story, like Gone with the Wind, or a tragic love story like Romeo and Juliette, but it isn't a Romance. Sweeney Todd's daughter gets a happy ending, but the story, as told through the musical Sweeney Todd: the Demon Barber of Fleet Street, is a tragedy with a basically decent man descending to murderous revenge.
Fashions change. Fifty years ago a Cozy almost guaranteed a village full of picturesque cottages and eccentric characters. Now you can expect a) a hobby; b) recipes; c) a pet who helps solve the mystery; or d) all of the above. What is considered graphic violence and/or sex has changed. The basic constraints on a Cozy remain the same.
Romance may be same sex or cross-species (if Science Fiction or Fantasy). I think you could even make a good argument for including the Bromance in the genre. But the protagonists always get a HEA. That's what makes a romance a Romance.