The device works well with all of my USB A ports (it comes with a very
thick USB A to A cable)
That's one classic issue with the type-A port used on a USB device. Type-A ports are designated for USB hosts. You were using illegal A-A cables to connect it. Fine.
Now you are using a Type-A to Type-C cable. This cable is designed to connect USB host (type-A) to Type-C device. Therefore it has HOST SIGNATURE on CC pins in the cable's overmold. Your new Type-C is also a host, so you are attempting a host-host connection, and this fails for obvious reason.
To get your SSD enclosure to work with Type-C host, you need a Type-C to Type-A Female adapter, aka "OTG adapter.
This adapter will have DEVICE signature on Type-C end, and FEMALE Type-A port, which will act just as an ordinary PC port. Then use your illegal A-A cable.
There is no single-piece Type-C cable which will connect your illegal device with Type-C port, unless you do a serious overmold "surgery" and switch the embedded pull-up to a 5.1k pull down on one of CC pins.
AFAIK, even non-removable / fixed drives (as in, its SCSI Removable Medium Bit is 0) can be ejected (as in, "safely removed") in Windows, as long as Windows finds the drive to be "external" / "de facto removable" (e.g. the connection type being USB).
(Note: I'm referring to the system tray icon only. It is normal that you don't see any eject button in Windows Explorer when RMB is 0, even when you have only one single partition / volume on the drive.)
Most if not all HDD/SSD enclosures have SCSI RMB being 0, whereas thumb drives normally have SCSI RMB being 1, and for the record, RMB is not something that can be toggled by the user like SCSI Write Cache Enabled (WCE), as its not part of a mode page. (But certainly, you can change it by flashing a modified firmware or so, if you can.)
So it's more likely that the issue is some sort of hiccup in the specific installation you are using, which is not a rare thing when it comes to Windows. It can be due to some registry crap or bug in some vendor-made drivers. The bottom line is, I don't think the enclosure is the reason. I am not aware that one needs to report anything explicitly for Windows to identify it as "de facto removable".
I have some not so convenient alternatives for you. You can go to the Settings -> Bluetooth & devices -> Devices and see if your enclosures are listed there and if so, see if you can "remove" it there. AFAIK it is equivalent to eject / safely remove.
Alternatively you can set the drive to Offline in Disk Management:
Which is even less convenient though, since you will need to set it to Online when you have the drive attached again.
I do NOT recommend changing the Removal policy of the drive to Quick removal, especially if it is some SATA SSD with volatile write cache in it, since it would significantly hurt write performance and might even cause greater write amplification. It might be more tolerable when your drive is an NVMe one though, especially if it's DRAM-less anyway.
(Also, in my experience, Quick removal does not always guarantee WCE to be set to 0. Sometimes the bit can be "out of sync" from the policy, but it could be some old-time bug that got fixed at some point.)
Best Answer
Your enclosure is NVMe, but your disk is not.
See the WD Blue 3D SSD (M.2) Review, where it's said:
Your SSD is not NVMe, so doesn't work inside that enclosure. It can work at SATA speed, which is 6 Gbps, but not at NVMe speeds.
M.2 is a form factor, so only detailing the shape of the disk. It does not relate to its technology, which can be either SATA or NVMe.