Best $500 System?

Bet those Reference 20s sound good!
Too bad there isn't a digital audio out of your CD player. Often a digital coax output is nestled among the component connections. The nice thing about the NuForce/Pioneer/DVD/CD system is that it's all new, with current technology and warranties. 
If you do need to go the DVD player route, there are a wide variety of prices & capabilities. At the upper end,($45-80), the DVD players have higher video resolution and streaming capability (if she wants to take advantage of the video capabilities),and still be below $500 total.
FYI (for down the road)-Last Christmas season, authorized dealers were running the compact Pioneer SW-8 MK2 subwoofer for $99 on Amazon. It's  a great match with the Pioneer bookshelf system. From your first post, it doesn't sound like she is focused on bass performance, but when you add the sub, tuned to a barely perceptible level, the combo becomes ridiculously good from top to bottom. In addition to a line level subwoofer out, the NuForce has an analog input so you can run a turntable at a later date , (with an outboard phono preamp). You can also use your CD connected to the analog inputs, without running it through the NuForce's DAC if you are really trying to save money.
Parts Express has a 24" steel stand for $55 pair that work well with the Pioneers.

+1 on the route proposed by returntomusic. A $299 amp/dac is a good start. I have listened owned the Pioneer speakers and they make for a great start within the context of a $500 system. Use any cd/dvd player as transport. Speakers and trnsport can always be upgraded as funds and desire dictates. 

Take a look at the Yamaha MCR - 040 mini system.

It is surprisingly good to start with - but with some simple tweaks it can be made to sound much better...
- install some reasonable speaker cables - small gauge
- replace the power cord

That's it!

I tried several boom boxes for outdoors music, but they all sounded crappy. I then looked at the various "music boxes" like Sonos but was underwhelmed.

Then I found the Yamaha and started with the tweaks above.

The speakers are surprisingly good for this type of unit, but the electronics are very good.

CD's, USB, iPod AND FM Radio

As long as you don't want to play it at really high volumes it is a good sounding unit with nice imaging.

And well within the budget - even with the tweaks

You can always upgrade the speakers later, but I've tried a couple of pairs I had lying around and the ones that came with the unit outperformed them both.

Something to consider :-)

Sorry about the duplicate entry above.
To get a better idea of how good the NuForce DDA-120 is, check out the rave Sound Stage review when it first came out in 2015, at a list price of $699!  (The Absolute Sound recommended it then too at $699, and continues to do so in 2017)
It seems to me that it was NADs de facto lowering of the list price on the D-3020 from $499 to the widely available street price of $399, that started a big price correction in this category (PS Sprout, etc)
NuForce  cut the list for the DDA -120 from $699 to $499. 
The sale at $299 is just crazy good, great time to be a budget audiophile!

A Play Station 1 is a must as your CD player.  Can find one for $20.00.
This gives you $480.00 for an amp, speakers and cables.  I would look for an NAD C320BEE Integrated for around $200.00 used and a set of Q Acoustics 3020 speakers for around $175.00 used.  This still leaves you $105.00 for decent cables.  This would make a great sounding $500.00 system.  Trust me.