Unless the form has been horribly butchered, that should still work, since the browser will select from the dropdown as you type, so long as the dropdown is focused.
If you implement this, please make sure any auto-refocus has visual clues, e.g. looking like a single input field, and behaving like a single field (a TAB moves to the next logical form field).
When inputting license keys in programs, and there were dashes in them, it was always fun encountering an auto-focus across split textfields where you didn't expect it, so you accidentally skipped a field and had to redo it.
But still dropdown help users to remember the date layout (MM/YY) and also it is useful in cases where it is not intuitive that you can type.
Also, expiry date is always somewhere around 5 to 10 years so not a lot of scroll.
You have to take your hand to mouse, do 4 clicks, then go back to keyboard. Not the end of the world, but it's annoying and is bad for people with. e. g. RSI.
And to remind users about date layout, you can just put MM/YYYY or MM/YY as a placeholder. Of course, if you care about your users and want to go above and beyond, you can even support both MM/YY and MM/YYYY – it's not that hard!