I think you need to stay focused on what you can get that will reasonably do the job, and less on expanding your needs and adding gadgets.
One amp will easily drive two pair of speakers. You need to have two rooms, one up and one down. That is a pair of speakers in each room. That amp will allow you to select upstairs-only, downstairs-only, or both.
That is your most basic setup; easy and cheap to accomplish.
If you want 6 speakers, three stereo pairs. That gets complicated.
Before we go any farther, GIVE US A NUMBER FOR YOUR BUDGET. 'Budget end' could mean anything.
Next...requirements.
Do you need a remote control (a common infra-red remote), or will physically having to turn the amp up and down by turning a knob be sufficient?
Do you need independent level control? Will you ever need the upstairs and downstairs volume to be different, or will you be able to reach a fair compromise with a single control?
How big are the rooms?
What is the noise level like? Is it more like a restaurant or more like a bar?
What is your intend? Is it to produce pleasant background music that people can enjoy but generally be unaware of...you know...background music. Or, do you intend the music to be more dominant than that? Example, in a cafe or restaurant, the music tends to be background to set the mood. In a bar, the music tends to be more foreground to stimulate and motivate people (usually to motivate them to drink more).
And speaking of a bar, do you serve alcohol? The presents of alcohol tends to raise the noise level even in modest cafes.
Realistically, your minimum budget should be Amp (£120), Speaker Set 1 (£100), Speaker Set 2 (£100), plus speaker wire for a Total of about £350.
To some extent you can control the sound by putting larger speakers in the loudest area, and smaller speakers in the quieter area.
Having a computer is nice because you can create your own play lists. You can play songs that all have a similar mood. Perhaps mellow and relaxing in the daytime and at diner, then for evenings, something more upbeat. But, unless you have great taste in music, playing random selections from you computer might not work so well. Though I think you still have a sound idea, and should continue to work in this direction.
Lastly, your diagrams or text-grams indicate a misperception. The [hi-fi] is the [amp].
Your options, within the limits of text graphics, are -
[PC]--->[HiFi Amp]=====>[speakers 1]------>[speakers 2]
or
[PC]--->[DAC]---->[HiFi Amp]=====>[speakers 1]------>[speakers 2]
I think you should start WITHOUT the DAC as it add considerably to the cost. It can always be added later if you are pleased with the audio system in your Sandwich Bar. How much the DAC will add to the sound quality will depend on the quality of your audio files. Also, note that likely you PC also has a CD player, that is another source of music that you may not have thought of.
As to the presences of a Speaker Switch, again, it depend on your amp and your speakers. If you never have more than two sets of speakers running, you can have as many speakers as you want. But the total Per Channel load seen by the amp should not drop below 4 ohms.
I say start with a basic amp and two pair of 8 ohm speakers, and build from there. Remember, with a basic amp, you can independently switch the two sets of speakers on and off.
It is probably even possible to integrate a small PA system into the stereo system for a very low price. Don't know if you need that, but it can be done.
For now, one thing at a time.
Bottom line, if you want to do this right, £300 to £350 is your basic start up cost.
Now, you can buy some ultra-cheap discount store stereo or boom-box, and it might work. But the quality is never going to match the systems I'm suggesting. Cheap discount store amps are going to be low power and high distortion.
So, if you can work in that budget, and expand your requirements and intended use, then I will make specific suggestions.
If your requirements are lower than I anticipated, we might even be able to trim the budget back...maybe...possibly.
Steve/bluewizard