If I want a fast backup I use the Ethernet cable.
The Ethernet ports are not enabled when you configure the Time Capsule to "join" the network using wireless only. You can only backup using wireless, whether you have Ethernet connected or not. Apparently, the "support" folks did not tell you about this.
I just do not see why this setup would take longer.
You are kidding, correct?
Gigabit Ethernet is 1,000 Mbps.
With a fantastic wireless connection, you might get 200-250 Mbps. Most users see about 100-130 Mbps.
As for why your Time Capsule setup will take twice as long as normal wireless .....
Backups have to make a wireless "double hop". Hop one is from your Mac to the wireless router. Hop two is from the wireless router back to the Time Capsule. Two hops = double the time.
Since you can only use wireless, your actual backup speed will be in the range of 100-125 Mbps, maximum. When you try your first backup on a Mac, it will likely take days, not a few hours as it should.
Since you have another hard drive attached to the Time Capsule, any copies of transfers to this drive will be exceedingly slow as well.
Unfortunately, the support folks did you no favors by providing the steps to this type of installation.
At least, they could have told you about the downsides.....which can all be avoided by connecting your Time Capsule to your modem/router using an Ethernet cable as the Setup Guide recommends.
thought that another reason to have and use a TC is to be able to use USB peripherals connected to the TC in wi-fi, like printers. This was the center of my question to start with.
Easy to do if the Time Capsule is connected to your modem/router using an Ethernet cable.
If you will, let us know how long that first backup of your Mac takes using your setup.