Some thoughts...
Your four-pair wiring is definitely not rated for data since 70's-grade copper was 10+ years before high speed data communications over consumer-grade twisted pair was even seriously considered. (10BaseT wasn't standardized until 1990, before that Ethernet was Co-ax and AUI).
Wiring houses with physical lines is IMO a losing proposition unless you need NSA-level security, run commercial-grade gear like some of my friends with equipment racks or you have performance requirements beyond what wireless can deliver. As you noted, it's also a PITA running wire in most homes since there are few good places to run cables and you really need construction-graded cable (plenum-rated) to meet code anyway. Also, frankly, the number of devices that even include a physical network port is dropping steadily.
I'd start with a high quality wifi hub/router in a good spot in your home. Don't be cheap here, spend time doing research on quality and performance with your target set a notch or two above your current requirements. Then you can use wifi extenders to fill in any coverage gaps. Modern 802.11ac 5GHz Wifi will cruise at 300+Mbit/sec to each endpoint which is pretty darn fast for most home use. If a critical device that you need to accommodate is not Wifi-capable then upgrading it to a wireless version would be easier than running wire for it -- or you can relocate the device to a place where it can directly plug into your router, of course.
Since your entrance seems to be in an awkward location and you have plenty of cable available, a compromise might be to run a single hardline from your entry point to the optimum location for the router -- but after that clean out the extra wiring from your home and enjoy.
Source: 25+ years in IT infrastructure, a home setup that has a lot less spaghetti than it used to and way too much time running wires through dirty spaces in the early 90's.
Good luck!
PS: If you're feeling particularly geeky, I am a huge-fan of Pi-Hole for network-wide ad blocking too!
One consideration:
If the OP or his children are gamers at all, they'll want a wired connection to whatever device on which they're gaming, PC or console. No matter how good your wireless is, it will suffer from significant latency compared to a hard-wired connection. Wireless is fine for turn-based or exploration-style games, but not at all for anything involving fast movement, such as shooters, racing games, or sports games.