Advantages of a tube CD player over an SS player?

No advantage.

I have owned an Ayon CD1s --- not the 07... and it is a fine sounding, rich, lively CD player well worth the used price.

When inexpensive --- say a $1000-$2500 CD players are compared it is likely that, in general, the tube players will be a tad more rich and lush sounding and perhaps a little less strident (though there will be some exceptions). Once you get to the 3K, 4K... 10K range, sonic differences between CD players will exist but not based upon tube vs. SS at that point. The point is cheap SS electronics are a little thinner sounding and maybe a tad rougher edged than cheap tube sound. Once one transcends the "cheap" range, the SS flaws are much reduced and tube vs. SS (on that basis) becomes a non issue.

last point... tube electronics are tremendously less reliable than are SS electronics. Keep that one in mind. I have owned both and over time favor SS --- blown tubes (and circuits that are damaged when tubes blow) are a pain. That said, the tubes in CD players are far more reliable than amplifier tubes --- so that with CD players reliability is much less an issue. Amps... that is a different story.

Sunnyjim, it's not the tubes that prevent "glare," it's the design of the master clock to prevent jitter. Poor jitter control is the primary cause of that "glare" sound produced by CDs.

I'd agree with what Charles1dad stated, which includes the quality and design of the DAC circuitry.

There are "colored" tube players and there are exceptionally honest and pure sounding tube players. SS players can be colored just as much, just in a different direction/characteristics. I've heard excellent examples of both and poor versions of either. I'd focus on quality of analog circuit, power supply and designer talent/ implementation.

It is true that some tube CD players just contain a tube buffer stage. A buffer is an additional circuit that the signal passes through that has no gain or amplification. If that is the case, then adding a separate tube buffer will accomplish the same thing, but requires more interconnects which will also affect the end result. IMHO, if adding a tube buffer results in a more pleasing sound, it's just masking a synergy problem somewhere else in the system.

As far as the Ayon CD-07, it appears that this is not a CD player with an added tube buffer. Their description of the player states Class A tube output stage and no buffers are used in the signal path.

I also agree with Charles1dad's comments, and both of these are excellent players. Provided there is good synergy in the rest of the system, either player should sound good, although likely a little different. My experience with another Ayre player and reviews of the CX-7 is that the sound is not anything close to being laid back. So, if your system is somewhat laid back, the Ayre could definitely add more of a lively sound, however, in another system, as in my experience, it may not suit someone's personal preference.