Good instinct asking before buying. One suggestion from watching a lot of people start this: separate the two jobs in your list. Getting photos off the phones automatically and keeping safe copies is one job, and a NAS is good at it.
Browsing and finding photos again is a different job, and it is where self-hosted setups most often disappoint beginners, because the polish of Google Photos search is hard to replicate. Plan the first job on the NAS from day one and treat the second as swappable. Immich is the strongest open-source answer for both, just budget real time for updates.
Since you asked for open source only, a short disclosed aside: I make a paid, closed-source Mac app in this space, and the honest summary is that if any machine in the house is an Apple Silicon Mac, native no-server options for the search side exist, PhotoCore among them.
For the stack you described, Immich on the NAS is the mainstream path.
The article is right about the money and misses what people are actually buying. Nobody self-hosts photos to save money. They do it to stop renting access to their own files.
The honest accounting has three columns: cloud costs money forever, self-hosting costs time forever, and both are fine trades when you know you are making them.
The option that rarely comes up in these debates is the boring middle: local-first apps on hardware you already own. No monthly fee, no server to patch, files in plain folders you control.
That covers organising, searching and keeping the archive yours. It does not cover off-site copies or phone auto-upload, so most people still pair it with one dumb storage target. Disclosure: I build a paid Mac app in that local-first category, which is why I notice the gap. For beginners my advice is to start local-first and add a server only when a concrete need forces it. That way nobody ends up as the person who quit self-hosting.