In all fairness, you are confused about what you want and what you are expecting.
Speakers rated to 60 watts are pretty weak, but 20 or 30 watts to a speaker designed for 60 watts is still a lot of power and can push enough volume to be more than what most people find 'comfortable' listening.
So, you are worrying about something you shouldn't instead of worrying about how to get the most from what you have purchased.
1. A speaker selector allows you to use a SINGLE stereo amplifier to drive multiple stereo speaker pairs. Typically it allows you to turn each pair on and off independently.
2. It should be wired with the amplifier feeding the input, and ONE speaker on each output.
3. The audio will be divided properly and appropriately among all speakers so as to not cause damage.
4. You may find it easier and cheaper to use a multi-channel A/V receiver and put it into 7-channel stereo and have all the speakers playing at once as well (this works).
5. You don't get individual speaker pair on/off control doing #4.
If you are looking for some sort of advanced level of control, then it is far more common to put in individual volume controls in each room of the house that has speakers, and wire them to a single amplifier, or to a multi-channel amplifier to control volume.
There really is a hundred different ways to do multi-speaker distributed audio setups, but it is not really clear what your final goal is and what (specifically) your gear is.
I would DEFINITELY not be wiring speakers together and trying to get additional power from things or anything like that. It's a rookie mistake and will only cause you problems in the long run. Most likely to the tune of damaging your equipment.
Get the speakers unwired.
Wire each of the 6 speakers to the 6 outputs of the speaker selector.
Run ONE stereo channel out of the amplifier into the speaker selector.
Use without fear of blowing your equipment up.