What did Synology support tell you? I would expect you could reinstall the OS from scratch (equivalent of Windows Safe mode) while still protecting the data partitions.
As noted downthread, it'd be a PITA to try and run Linux mode in Windows. Best way is to find someone who is Linux literate that can mount the drive in a native Linux environment and move the data. As a RAID 1, I agree it should all be there on one drive.
I am not a tech expert and there may be risk of data loss in doing this. This is more of "I've tried everything" idea.
Your RAID controller may have gone bad. The linux advice is the right advice, but you need to disconnect one of the drives from the raid controller, and connect it directly to the motherboard, and see if you can get access to it that way. (Do not hot-swap it, change connections where everything is off.)
In theory, RAID 1 with two drives is just a exact copy of a single drive. In theory.
I pulled both drives and have one plugged into an internal port on my PC. Between google and a friend I figured out how to get a Linux module up and running, but the Linux module won't "mount" the partition on that drive that has the data on it. I am emphatically to the point of it making sense to pay someone to deal with it.
Sorry didn't know you already did that step, thought you only booted into linux not rewired.
I haven't done this in a while, but I think this distribution will have the tools you need to mount the disk.
Basically you'd start your computer from the USB drive and run gparted to see if you can see the disk. I can't quite remember if RAID 1 has any special sauce other than writing twice, but I believe each drive should be visible.
Feel free to email me - I love a tech mystery or two.
if the need allows that. I think he is very good and very responsive. If you like, let me know and I'll introduce you.