Time Machine Initial Backup Larger Than Hard Drive Size

Apple's inexplicable decision to discontinue Time Capsule along with the rest of their AirPort line of products left us with a lot of bad choices. The directly connected USB hard disk drive is the simplest, but not convenient for the reasons you describe. You do have to remember to plug it in once in a while, however, TM continues to create "local snapshots" that are fully restorable complete system backups, all the time. The obvious limitation is that they are good for only about 24 hours, and if the Mac breaks before it flushes them to an external disk those intervening backups are lost. So plugging them in "once in a while" is ok. You just have to remember to do that. If you don't, TM will remind you after ten days has elapsed.

The other bad choices involve using those old discontinued devices, which are readily available on the used market. As old as they are I have never had one fail, but if that were to ever happen they are repairable — with difficulty, and within limits. Those devices can be any combination of Apple AirPort Time Capsules (all models ever produced since the original one), Apple AirPort Extreme Base Stations (the most recent, "tall" models) including any number of USB hard disk drives connected to their USB ports. More than one USB hard disk drive will require the use of a powered USB hub. I have used many with nonpowered hubs and they work fine, but that should be considered an exception to the rule since the TC's USB port power is limited. With those devices I have never encountered the slightest problem to back up or restore, ever.

You wrote that the Synology is using two 12 TB HDDs. I assume you intend that NAS to be used for purposes other than Time Machine, because such a copious amount is well in excess of what Time Machine requires. If the source volume has 1 TB capacity then a 2 TB backup disk will be adequate, though a 3 or 4 TB will be much more comfortable. They will all be about the same price and inexpensive enough to have more than one. When one fails, throw it out. Be sure to encrypt backups, for that reason.

Apple says,

• Use a storage device with at least twice the storage capacity of your Mac. For example, if your Mac has 1TB of storage, your backup disk should ideally have at least 2TB of storage. If Time Machine needs more storage space, your Mac will let you know.

As far as non-Apple NAS devices go I encourage you to participate in a Synology product user forum. Apple simply won't care about Synology, so it will be incumbent upon you to keep on top of software / firmware updates from Synology and for them to address bugs when they arise — which they will, perhaps with every incremental macOS update. I happen to know they have been very poor keeping up with them. It's a full time job and I got the impression they are insufficiently motivated to do that. They sell you a box, they get your money, the box works for the duration of its warranty, and their job is done. Time Machine support is a very distant afterthought.

Is that something particular to Synology? I've never had a problem using QNAP NAS as a TM target, and there is built-in TM support (slightly better using QNAP's Hybrid Backup Sync application on the NAS instead of the built-in backup tool - but either works). Plus the support for SMB3 and newer TLS versions. I'd be surprised if Synology didn't also have TM support.

For the OP - silly question maybe - is it possible you have external hard drives attached and TM is trying to back up your computer and the external drives too? And how did you set up the Synology to support backups (like I said above, I am a QNAP user)? Is there a back-up and sync program from Synology that has a TM target built into it, or how does it work?

If you purchased that Synology NAS to serve as a Time Machine backup, and the purchase is recent enough to return it for a full cash refund, then do that. It will never work.