quoted

### btrfs fi df /

Data, RAID1: total=4.49TiB, used=2.90TiB

There are about 1.5TB of available space in existing block groups, so
those will be filled in first.  Block group allocations will start on
sdc2 after the existing block groups are filled.

Ok! This is what I didn't know.
Now it's clear why Btrfs is not using /dev/sdc2 currently.
New blocks would be put on /dev/sdc2, but only after
currently allocated blocks got filled.

Unfortunately balance is the only way to redistribute the existing data
onto new drives, so if you don't want to run balance, then you'll just
have to wait until sdc2 fills in naturally.

Your answer above solved my problem.
Thank you.