No from Kodi cannot see any folder
It won’t work.
Kodi on your OSMC box is trying to negotiate hmac-sha1 which is not currently supported on your Synology box (probably because it’s not considered a very secure algorithm any more).
I keep telling you this - the problem is quite clear from the logs you posted …
Go and read this thread but be warned that fiddling with the encryption protocols is not recommended unless you know what you’re doing and, in this case, the Synology box isn’t providing services to The Internet (E.g., you’re not accessing it across The Internet).
If you are not accessing the Synology box across The Internet (E.g., if that and your OSMC box are on your LAN at home) then don’t use SFTP, use NFS (or CIFS) instead.
The server on the Synology box saying “I can provide these algorithms”: hmac-sha2-256,hmac-sha2-256-etm@openssh.com,hmac-sha2-512,hmac-sha2-512-etm@openssh.com,umac-128-etm@openssh.com,umac-128@openssh.com
The client on the OSMC box (Kodi) is saying “I only support these algorithms”: hmac-sha1
Until you can get your server also to offer hmac-sha1, or your client (Kodi) to accept one or more of hmac-sha2-256,hmac-sha2-256-etm@openssh.com,hmac-sha2-512,hmac-sha2-512-etm@openssh.com,umac-128-etm@openssh.com,umac-128@openssh.com, the two will not talk to each-other.
As you are able to access it from the command line you might could go down the path and mount the external filesystem via the kernel driver instead of from Kodi.
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Incidentally:
So your server is xxxxxxxx.synology.me (obviously obfuscated 
).
Looking at the image you posted, you’ve entered a source of xxxxxxxx.synlogy.me - should be ...synology.me - you’ve apparently entered the wrong host name …
… which won’t help either …
… but seriously, go and sort out your key exchange algorithms (or use kernel mounts as @fzinken suggested; or use NFS or CIFS if you don’t need encryption; or …) - that’s where your problem is.
So, is there no way to add sources from the OSMC skin?
Videos -> Files -> Add new source (or something like that, I have Polish language).
I am not sure what issue you have. But generally the OSMC skin offers easy ways to add new sources as @ooZee shown in his reply
The problem is that BROWSE doesn’t do anything like it does in the Kodi skin.
Care to be more specific? On my system Browse looks pretty much the same on Estuary and OSMC skins.