The same way the system allows you to type in 0213 without blurting out a big error the moment you type 02. Just don't validate the input while the user is still inputting it.
Secondly, remember this is for credit card expirations and not normal dates. You can probably make an assumption that 11/30 (November 2030) isn't currently valid input either. I have never seen an expiration date further than 10 years in the future. Although more research on the topic would be needed to figure out the true restrictions.
I don't know why the live demo is broken, or why they even need is since there is a live demo on the main page, but the usability for all cases strikes me as exceptional on the main page where it works as intended. I am not convinced that your cherry-picked restriction improves the usability.
Secondly, remember this is for credit card expirations and not normal dates. You can probably make an assumption that 11/30 (November 2030) isn't currently valid input either. I have never seen an expiration date further than 10 years in the future. Although more research on the topic would be needed to figure out the true restrictions.