Please be aware that unless you plan to move out in a week, you should get decent components that will last the lifetime of your home.
If you plan to use ONE source and have it shared across multiple rooms, you may get impedance matching volume controls and use them everywhere, then use a single amp, or a couple of amps to drive everything. That's the least expensive route.
Another option is to run separate channels of amplification for each room. Most amplifiers can handle a 4-ohm load, so that means two speakers per channel of amplification, and rooms will be stereo, so two channels of amplification per room.
You can pick up 12-channel amplifiers (6 rooms) for about $300 on eBay used with great quality.
ie:
https://www.ebay.com/itm/Sonance-SONAMP-1250-12-Channel-Rack-Amp-50Wx12-Good-condition-PreOwned/133411014917?epid=66797633&hash=item1f0fea8905:g:VNwAAOSw2PVevb-h
You can pick up 16 channel amplifiers (8 rooms) as well. Look around, get a good deal, make use of it.
These allow for individual audio selection for each room and power that is individually fed to each room. This should make for a longer lasting and more stable audio system.
Now, you say $250 for prewiring 4 rooms. Decent audio cable is 60 cents a foot, and it may be 50' per run, so $30 in cable, times 4 rooms, that's $120 in materials. So, $130 or so for the labor of running those rooms is SUPER cheap. Typically it would be at least $100 more to do that work.
I don't know what to say here. You will need speakers in all these rooms. $60 a pair minimum, plus installation costs. I would spend a bit more to get better speakers. Installation costs stay the same. You do need a way to control volume in each room, and I'm unaware of many units which offer remote volume control on the cheap. Volume controls are a much better option for single source volume control on the cheap.
A single source like an Amazon Echo. Hook it up to...
A single amplifier:
https://www.monoprice.com/product?p_id=18514
A audio splitter like this:
https://www.monoprice.com/product?p_id=38161
A bunch of cheap volume controls with impedance matching:
https://www.monoprice.com/product?p_id=38170
So, for a few hundred bucks you get audio with in-room volume control from a single source.
If you have 6 rooms (zones) of audio total, and you want multiple sources, and the ability to change things from within the room, then you have to spend the money to get a multi-source, multi-zone system.
Distribute and control music and other audio material to up to six stereo speaker zones using this 6‑Zone Home Audio Multizone Controller and Amplifier.This multizone controller is a 6x6 matrix
www.monoprice.com
or
Whole-house audio systems you can install yourself and control with keypads and smart devices. All HTD systems include a 30-day guarantee, 2-year warranty.
www.htd.com
I'm not sure how spending $1,100 or so on whole house audio is too much when you want that level of control. That's what it costs. Simple as that.