The advantage of doing this is having a more flexible and customizable solution. Not only is it a NAS but it can also run VMs and docker containers (letting you run various services like Plex video streaming, PiHole, personal 'dropbox', personal VPN, torrent, media ingest station, security camera recorder, home automation, periodic backups etc...). So when you want to run something 24/7 in the background you just start up an instance on there and it runs, rather than scrounging around for a machine to run it on.
You also get more hardware flexibility. You can stick in a 10Gbit network card, you can stick in a SAS HBA (letting you connect multiple 12 bay arrays of drives for example), you can use a SSD as a cache, you can talk to a UPS for safe shutdown, etc... None of this applies for little Linux SBCs like a Pi tho, they are under powered.
I do all of this on my QNAP NAS. It has a 4-core/8-thread AMD processor and 32GB of RAM, runs VMs and containers, has an add-in 10Gbps PCIe NIC (plus the built-in dual 2.5Gbe NICs which can be bonded for 5Gbps throughput), dual 1TB NVMe SSDs for caching, connected to a UPS to manage automatic shutdown after lengthy power failures.
Synology vs QNAP: { pros, cons } ?
Eh, that would quickly devolve into a subjective debate. Back when I did the research I was leaning toward Synology as it was the one I was most familiar with, but after reading about support, software, etc. I felt QNAP was the better choice for me. And I have not been disappointed. Both are capable and have their fanboys, so I won't go there, and don't have a Synology to do A/B testing against. All I can say is I can highly recommend QNAP from experience, having owned 2 of them since 2015 (a TS-851 and a TS-673A).
{ HDD, eSATA enclosure }: which brand and model?
I ran Hitachi/HGST and Toshiba desktop 7200rpm drives in my NASes with great success. Not even NAS rated drives. I ran them for years, first the aforementioned HGST 500GB drives, then upgraded to HGST 1TB drives, then Toshiba 3TB drives in my QNAP TS-851 in 2015. Only with my latest QNAP TS-673A did I spring for WD Gold Enterprise class drives.
As to the eSATA enclosures, I ran Sans Digital (two 4-bay and one 8-bay) enclosures using dual-port eSATA cards. I still have them collecting dust in my garage, LOL. OS was base Debian or Ubuntu Linux running mdraid, all custom configured (no NAS-based distributions like Unraid or Freenas).
I am of the opinion that a "personal home" NAS should never be turned on 24 hours a day, it simply doesn't make sense.
Respectfully, I would strongly disagree, and would argue the exact opposite -- it doesn't make sense NOT to leave it on, as you would any server (which is what it is). And as far as RAID goes, it isn't perfect by any means and as mentioned, is not a substitute for backups. But it works damn well and is highly reliable if you know what you're doing. I do prefer software RAID (mdraid) to hardware RAID.
My NAS never powers down (well, short of planned maintenance or extended power outages). It serves many purposes, including media server (both to the internal network and when I'm travelling), backups (both from internal and external), personal cloud storage, internal file shares for multiple systems, etc. Even if I didn't require external access, I don't want to have to boot up a NAS when I need access. The wife would not be happy if she had to wait for me to boot it up so she could watch or listen to something.
As far as external access, I also have a pretty sophisticated home network (including 10Gbps network between the NAS and core workstations) and experience configuring networks securely, so there are multiple layers of security between the internet and my NAS. At the risk of jynxing myself, neither the NAS nor my home network have ever been breached and I've been running internet-facing servers for 20+ years, going back to dial-up (yes, I ran servers on dedicated phone lines).
Powering up and down a NAS is a pretty bad idea. If you're going to do that, may as well just use a big USB-attached external drive.