Well, it absolutely won't! This situation is very familiar to me here and Powerline is my go-to solution.
Stick to 2.4 GHz (congestion will not be a problem

) and, if you're using TP-Link units, get familiar with their utility for device setting, which opens up some useful features (power adjustment, cloning settings, naming APs, ...). If the router has 5 GHz, disable it for improve roaming between the router and the other APs.
Doors between rooms provide a route through the walls for the Wi-Fi, but that's usually a disadvantage: place APs where they are shielded from covering adjacent areas (horizontally), at least initially. If you have overspill through a doorway, it can wreak havoc in the area it's spilling into.
If it starts getting too expensive for full cover, consider explaining that plug-in units can be moved seasonally – if there's room that's only used in summer, for example – and that additional APs can be added later, to spread the cost. Additionally, fixed clients (e.g., television, Firestick, sat. box) can be connected to a Powerline adapter by Ethernet cable, which may simplify things, and fewer Wi-Fi clients is always a good thing. The same goes for printers, if they have Ethernet ports. Disable the Wi-Fi on hard-wired clients.