If you already have an old laptop, and the battery will hold a charge for even 20 minutes, then it might be worthwhile to just give it a whirl. If you have it tucked away with the lid closed, it's possible that it'll run slightly warmer than it normally would, when running 24/7. And if you do find that expandability is too constricted, then upgrading to newer hardware and moving the HDD/SSD over should be easy enough.

IME, mortality bias means that if an old laptop is still functional a decade after its manufacture, then it's likely to keep ticking when in service as a 24/7 machine. The things that permanently break hardware are thermal stress, capacitors charging up and down, and other factors which are alleviated when a machine is just left powered.

Laptop all the way.

The mini PC and UPS are separate systems with no integration. Trying to get them talking so that your mini PC can respond to lower going down by saving things and shutting down cleanly is a lot more work than doing the same with a laptop.

Also a laptop you already own costs you nothing to get started with. If you add some external storage to it and then one day add a server you can simply move that external storage over to a new server. So if you do end up doing self hosting or a home lab you will well on the way.

You need to keep in mind that most laptops are not going to have 2 ethernet ports(or even 1 these days). You will need 1 port to connect to your modem and 1 port to provide the network to an AP. All ports aren't created equal. If you want good speeds you want to find something with 2 good ethernet ports. Your internet speed will determine what is good enough for you port wise. If you are fine with pretty slow speeds then a laptop can work and you can use a usb to ethernet adapter but that's a bit janky. If you intend to use the built in wifi card for the AP just know it won't be nearly as good as a dedicated AP. Check out the Lenovo ThinkCentre line. It has some good options with decent ports. As for a UPS you can sometimes find something better by looking at "Solar Power Generators" instead. They're basically just batteries with an inverter and outlets on them. And they can recharge from solar which is why they have the name. But a lot of times they're the better bang for your buck than a normal UPS. And if you got a few solar panels you could hook them up to it. Which it sounds like might be useful to you in general.

Laptop: cheaper and easier, but keeping the battery charged constantly will degrade it (no matter how many battery protection features it has), and upgradability is somewhat limited.

Mini PC + UPS: more upfront cost, some elbow grease required to integrate power monitoring (but it's not difficult), but more flexibility to expand, and the UPS can provide power to other devices like an access point. Also, the UPS battery uses a different chemistry and is made for that specific purpose, meaning it'll take longer to degrade.

You absolutely need a UPS, even if you have a laptop, power going in and out and being out of voltage is not great even for a laptop