Handoff is handled by the client, not the access points, and the client tends to do a bad job. My iPhone, for example, is dumb enough to stay on a dying connection for like 30 seconds as I walk right next to another AP it could connect to, so I keep having to force it to rescan. My parents' house is a hostile environment for wifi signals themselves due to materials in the walls, so there are many APs, and the problem is extremely apparent there. AirPort Express totally failed the test there. Same with my college housing a few years back, except they had higher-end equipment.
If you buy special wifi equipment with fancy tricks to help with this, it works a lot better (but still not flawlessly). Seems like those employ hacks to get around the deficiency in the wifi standard, like spoofing values or forcing clients to disconnect at a higher threshold, and the latter requires some tuning. Standard equipment like AirPort or whatever modem/router/wifi combo your cable company installs doesn't do the job.
Another issue is interference between APs. Pros are careful to avoid this, then they tell everyone not to set up their own APs. Wireless mesh setups avoid this automatically, though TBH I'm not sure exactly how they do it.
As a result, if my requirement is that I have at least 1 Mbit/s at all times, in my experience LTE has been better than wifi even in buildings with pro wifi setups. I've almost never seen my LTE fail, but wifi always has dead zones.
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It's possible. BT does support networking, I think IP at the lowest. BT 5 theoretically supports 48 Mbit/s. That's way more than enough for most. If I can get even 4 Mbit/s with Bluetooth but have perfect connectivity and easier setup, I'd gladly take that tradeoff.
Such as?
I have had a great experience with my Ubiquiti Unifi setup. 24 port PoE switch, 2 Pro APs and their Security Gateway Router. It is also scaleable so I can run a line and add another AP if I feel I need one (like in garage or backyard, etc).
With a POE switch you do not need a power outlet in the ceiling for each access point (i.e. PoE "Power over Ethernet"). Also, with ethernet ran to each access point it means you get full gigabit backhaul instead of these extenders that reduce speed as they need to hop from one to another. To my knowledge this is not much different than a mesh network, just with gigabit wired backhaul.
You can literally look at the Unifi management software and see a device (like an iPhone) move connection from one access point to another seamlessly.