This is usual and proper procedure, and as such that is fully OK.

Maybe usual and proper in Jiangsu province! But what about elsewhere in China?

It's a problem if they ask you to teach before all this paperwork is completed. There are many, many pitfalls and traps you could run into if proper procedure is not followed.

Proper procedure elsewhere in China might well be deemed improper procedure in Jiangsu!

And I have to say it again -Chinese law should be fully respected and followed in this matter!

And I have to say that other parts of China may have different rules and regulations that don't necessarily follow Chinese law to the letter.

For example, I heard the other day from an FT whom I trust very much that his local PSB office has just cancelled his L visa on its expiry date, and replaced it with a Z visa and an RP. He had already been teaching at his new school for about two months. He's over the moon, he needn't go on a Hong Kong visa run!
However, it's not all good news - there is at least one part of China, an autonomous region, as it happens, where the Regional Education Department based in that region's capital requires a foreigner to produce a recent criminal background police check from his homeland, which is clean as regards cautions for serious crimes or indeed serious crimes having been committed, before he can get his FEC. But no FEC means no RP. However, don't quote me on that - there will probably be exceptions.