Didn't notice the replies here, until the latest one.
Wired connection: I haven't done that, because the system in the lounge - where a single Dot would be placed - is a stereo system that can be switched off. Using BT, when you turn off the system, the BT connection drops, and the Dot uses its own speaker automatically. Using a wired connection, when the stereo was off, I would have to remember to unplug the speaker wire from the Dot, otherwise I wouldn't hear any response from it.
Amazon Tap has only recently been available here in the UK, and certainly not when I bought my Dots.
How I am currently using Dots and Bluetooth: I bought a number of Pure Jongo devices - all of them fairly cheaply, on eBay. Best way to describe them is a cheap Sonos alternative. Each device - I have 3 speakers, and 2 "system adapters" (for plugging into existing stereos) is equiped with Bluetooth. In my house, each device is in a separate room, and paired with its own Dot, in the same room.
Here is the magic. The Jongo devices can be optionally configured so that whatever one is playing, the others also play, in synch with each other. In this configuration, only one BT device can connect to the network, but that connection can be to any of the Jongos on the network.
So:
1. I am in my lounge, I say "Alexa connect", and "Alexa Play...". Whatever the lounge Dot is playing comes out of the stereos in the lounge and my office, and the speakers in the kitchen, conservatory and bedroom.
2. I go upstairs to my office, and decide I want to play something else. I say "Alexa connect", and the network drops the lounge Dot connection, and allows the office Dot to connect and control the system.
It works like a charm, and means that I can have the same music or talk radio playing throughout the house.
The only minor problem is that, having discovered this and gotten it working; it appears that Pure have discontinued the Jongo range. I'm ok, as I have more than enough devices, but it's no longer a working solution I can share. (Although Jongo devices do still appear on eBay)