Connecting a single audio device to multiple outputs for whole house speaker system
EDIT:  This is to address the additional info the OP add to the question.  The distribution panel you linked to is straight forward and is essentially the same thing as the switch i referenced in my answer below.  You run speaker wire from your amplifier or receivers speaker output to the &quot;input from amplifier&quot; (the red inputs).
On the back of your amp you have speaker outputs, each speaker gets to wires,  the left has a (+ &amp; -) and the right has (+ &amp; -) .
These go to the red inputs box on the panel.
The left from the amp goes  to the (L + &amp; -)  and  right from the amp goes to the (R + &amp; -)  ALL connections are Positive to Positive, Negative to Negative.
Your existing speakers are wired  colored coded. As long as you use the same color for each terminal you are good. In your case you have the old terminal blocks with labels. In the photo the block is set up as follows,
Red is right +, Black is Right -. White is Left + and Green is Left -.
To wire each set of speakers you wire them to the terminals on the new panel just same.
(Red to right +, Black to Right -. White to Left + and Green to Left -.)
You need to get the colors the same on your new panel as they are on the old blocks because they all go to a volume controller and from the volume controller to the speaker in that + - configuration. If you get any backwards you will have reversed the polarity  of that speaker (s). (it needs to be positive to positive all the way from amp to panel to volume control to speaker.)
I would suggest that you contact the folks at Crutchfield  They are very knowledgeable and will be able to assist you. They can tell you if the panel you listed is the best option for what your system needs, plus they have amps.   Not affiliated just a customer.

Original Answer:

Connecting a single audio device to multiple 4-pole audio inputs for
whole house speaker system

While there are 4 wires in the one jacket of insulation it is not a &quot;4-pole audio inputs&quot; It two sets of wires for two speakers.  Each speaker takes only two wires, a +positive and a -negative. (there are speakers with 4 terminals for bi-amping that can use 2 or 4 when two different amplified signals are sent to the same speaker, you should look at the back of each speaker to make sure what kind they are. Bi-amp speakers are usually a more high end audiophile speaker.)

simply plug in an old phone with an audio app installed

First it is not that simple. Your speakers need to be supplied a powered signal and your phone in not capable of doing that.
There are several options to get music from your phone to your built in speakers, the simplest would be a small stand alone amplifier with RCA inputs and out ports for speakers. (you will need an adapter to go from the headphone jack to RCA.) There are hacks and/or adapters you can buy that you can add to a stereo to receive a bluetooth signal from your phone.
To get the most out of your music/speakers you should have DAC (Digital to Analog Converter) Many amps have them built in or you can get a stand alone unit.
You are not limited to a basic amp, you can use an old stereo receiver or a modern AVR (Audio Video Receiver) or even an old/new tube amp. You will want an Integrated amp unless you want to go with a pre amp/amp combo but that is more complicated then we need to get here.  All of these have RCA inputs to accept the adapter from your phone and speaker connection ports on the back.
You have quite a few speakers judging by the number of wires you have. If you only want to use one or two  pairs then an old stereo will work, an AVR will have more speaker outs, but maybe not as many as you need if want to use all your available speakers.
The system in your house must of  had a speaker switch box with speaker outputs to all of the speakers in your home. I have a Niles and it has connectors similar to the ones in your photo for 6 pairs of speakers. A speaker switch has the ability to select one pair of speaker, all of them or any combo the user desired, (volume is controlled by your amp/receiver.) You connect the speaker switch to your amps , receiver or AVR speaker outputs and it will send the split/send that signal to any speakers you select.
I suggest that you join and Audio forum were you can read many treads and have a back forth discussion with others.
stereophile
and
audiokarma  Are two but there are many.
I would suggest that you get a turntable and a good Integrated amp. Leave the iPhone for texting and making phone calls. :).