'a better CD amp' ..... honestly, there's stuff that is probably better (Denon DM41DAB, Denon CEOL N10 or N11), but nothing that would make me want to raid my music collection every time I returned home. The problem is all of it is lifestyle gear, so there are a lot of compromises to get the features in to a small box at the right price. I get the feeling then that the reviews wouldn't ever pitch this sort of stuff up against full sized Hi-Fi separates.
The biggest compromise is the amplification. Much like your Pioneer, the power amps that drive the speakers are based on a design called a switch mode power supply. This is much smaller and more energy efficient than a traditional 'transformer + large capacitor' Class AB power amp design.
They're frequently referred to as Class D or Digital amplifiers, and the inference is that 'digital' is a good thing. They can be. There are some eye-wateringly expensive power amplifiers by high-end Hi-Fi brands where the designers had carte blanche to make the best products. But it takes a fair chunk of cash to make a true Hi-Fi performance Class D amplifier, and there just isn't enough money in a typical mini Hi-Fi system to throw that much cash at just the amp stage when so many other things such a selling price or features are higher priorities. What we have in a typical mini Hi-Fi is a cost driven design.
There were- and still are- compact Hi-Fi amplifiers based on Class AB power amp designs. The brands Cyrus and Rega both make such products, but not the sort of amp/CD integrated product that would compete in features terms with mini Hi-Fis. To get something Class AB which fills that niche requires looking back to the 2000s. Both ARCAM and Linn had some cracking products.
For something that has to be the same width as your Pioneer, then the ARCAM Solo Mini is probably impossible to beat. New, they were £800. Used, they can be picked up for under £200.
Where space isn't at such a premium, but you still can't fit in full width Hi-Fi gear at 440mm wide, then the Linn Classik Music or Classik-T (yes, spelled with a K) are excellent products and just 320mm wide. It takes the performance up even further than the ARCAM and they're every bit as good as full sized separates. These were around £1000 new, so s/h prices are around £300-£350.
Both of these units have more real-world power at far lower distortion levels than any conventional mini Hi-Fis, but that's not the main attraction. It's the sound quality. The longer you live with the ARCAM or the Linn, the more you'll hear from your music. It's very seductive. You'll get to hear different layers in the mix, space, timing, detail without excessive brightness. Stuff will pop out of the mixes that you never heard before with your current CD/amp. A good system will make you want to grab disc after disc and just keep playing music all day and night just to keep getting that hit of discovery.
The only catch is that as the systems get better then the gulf in performance between CD or vinyl and lesser sources such as Bluetooth will get wider and wider. A Bluetooth adaptor can be added as a source. Where your phone supports it, you can improve BT performance by going for one of the aptX HD adaptors. They use a more-efficient CODEC so that a higher resolution signal can be passed.
Both of these amps will work a treat with the Diamond 220 speakers, but could easily take on a pair of higher-end bookshelf or floor-standing speakers at £1000 without batting an eyelid. Similarly, a Pro-Ject T1 Phono SB @ £299 upfront will sound great with either CD/amp, but you could upgrade the TT closer to a grand and still not phase either of them.
In reality you wouldn't change the TT so soon but you would go for a better cartridge after about a year or so. The standard Ortofon OM5e is decent enough as a starter cartridge and you'll have a lot of fun with it. Changing to an Ortofon 2M Red will bring some big improvements though. Smoother top end, better midrange, tighter and deeper bass. Well worth the £100 it costs to change.
The next change up would be to swap the Red 2M stylus for the Blue 2M stylus @ £125. TBH though, you'd be better off buying speaker stands or even upgrading the speakers first.