Crosstalk Solutions has a couple of videos on similar setups: Log Cabin Property and Greek Port. Good tips on how to design it using Ubiquiti's free tools too.
You'll need a point-to-multipoint setup with APs as rebroadcasters. Just having a powerful central antenna is no good because the antennas in the client devices are your limiting factor.
I have used Ubiquiti for my home network for years, and it has always worked great at what it promised to do for me. People's frustrations with it—in my experience—usually revolve around things they want it to do that it doesn't, and the fact that it is very unfriendly to hacks or jury-rigging. So you have to be very sure going in that it matches your requirements. It's like Apple gear.
Another option you could look at is TP-Link's Omada system, which is basically their clone of Unifi but has a wider variety of hardware options.
I also often see people using Ubiquiti switches and APs in combination with a pfSense firewall if they want more sophistication and configurability. If you're coming from Sophos and want to do sophisticated routing and filtering, you'll probably find the Unifi firewall lacking in features (but it works great for people who don't want to do that).
That is a connection to a roof-mounted TV aerial from back in the days when that was a thing. No, you can't use it for internet.
Some scoping questions:
Don't buy Cat8 cable, the router doesn't have the hardware to benefit from it. Save a buck and get this Cat6 cable instead.
What router are you replacing?
You're going to have trouble finding a new, up-to-date wifi router for $50. If you can afford to spend a bit more than that, look at the GL.inet Flint, which comes with OpenWrt out of the box. Otherwise look for previous-generation routers on Amazon.
Unifi APs can do wireless backhaul and the system has good VLAN support. I would assume Omada is the same but I don't know it well enough. I'd suggest a central U6-LR, wired if you can, and one or two of the UAP-AC-Mesh (which despite the name has no special meshing features) for infill.
I have nothing to add to the comments about double NAT, but I wanted to chime in to ask if you'd considered something like a GL.inet travel router? It's OpenWrt instead of pfSense but I think it can do anything you want, and it's cheaper than a Vault.
Would it be easy to set up an AP with the same SSID as my router?
Yes, definitely. u/leewhat's comment explains why you had trouble with the Mikrotik; a dedicated AP doesn't have a router mode you need to disable, so all you have to do is specify the same SSID and your devices will take care of connecting to the one with the strongest signal.
Since you have a 3-level house you may find you need two APs. As you have discovered with your existing router, wifi signals are usually shaped like a squashed dome; they have a better range horizontally than vertically.
LWT will sell you an AP11 for $163 ex GST. Anything cheaper than that, you'll have to trawl eBay or Gumtree. But why Aruba in particular? There's no shortage of second-hand Unifi AP-ACs (e.g.), you can find the U6+ on staticice.com.au for around $170 (e.g.), and the Omada EAP610 goes for $140.
As other commenters have said, this is what a wireless access point (AP) does. You'll put the AP in the attic, connect it back to your router with one of the Ethernet cables, and it will broadcast a wifi network throughout your house. Examples that suit most cases are the TP-Link Omada EAP670 or the Ubiquiti Unifi U6 Professional.
How big is your house, and how many levels? This will affect the best option for you.
Return it. Buy a Cat6 cable and an unmanaged switch. Ethernet splitters are a scam, and Cat 8 is a cable for special applications in big data centres and nothing you own can take advantage of the ways it differs from regular Cat 6 unless you're Bill Gates.