In general seamless routing is kind of overrated. I enjoy pretty danged seamless routing at home with my 3 different access points with clients handling switch over and its seamless on all my devices with no issues with sticky clients. That's with 6 different hardware setups and 4 different operating systems going on between them. Streaming, page loads, downloads, video chat, etc all fine.
Frankly if you plan to run a wireless router with the radios turned off, I'd just look at a plain wired router. Some of the nicer ones aren't all that expensive (same price range as a new AC router) and they typically have a lot more options for you in regards to fire wall rules, routing performance, etc. At least if you are talking the price range of an AC router ($100-150). Possibly better VPN performance too.
From what it sounds like you are running, most of that stuff is not being routed (unless it is going over the WAN) and at most its doing some minor DHCP work for you. So...probably nothing. If you are streaming a lot of that stuff through the router to the internet a lot of the time, then yes, a newer/better router would likely be of great benefit, especially as it would likely have better QoS management than an older router would.