Yes, I can confirm that is the design behaviour on the B737.
Why would you want it any other way?
However, disappointingly, the IAS/Mach changeover doesn't occur approaching FL300. I wish it did.
It occurs around FL260. This means it will change from an "appropriate" climb speed of say 280 KIAS, to an "inappropriate" climb mach of M0.69 or thereabouts. Which, if let continue unnoticed, will cause considerable embarrassment at higher levels.
Edit: I just read your first link. My point exactly. This is just one of those 737-isms that keeps us on our toes.