A turntable that plays all types of music well

Hello,
I am going to say Origin Live. You can look at the website to see where your budget fits. I am really impressed with the owner. He has a strive towards perfection. Also, if you want to get a new arm they have great options. You are definitely set on your cart unless you want to ditch the phono stage all together. Just another option for you to look at. 

Clearthinker

That is what I am thinking as well. The TT itself can have a feature set that favors how the TT is used for (Example DJ, archiving, etc), however, I do not see the table itself favoring any genre of music. 

Weirdly, my Linn/Akito rig will not play polka...just won't...

Lol!

The two LPs just didn't have the same thing on offer for the cartridge to pick up?

I have a decent collection of 50s-70s jazz records. You can hear the sound quality change as they moved from tubes to transistor, from live to studio mixes, and from both to digital. Comparing the acoustic of a 1958 Doc Evans to a muddled digital cacophony is apparent once both the recording technology and the method of creation change.

I highly recommend the Thales TTT-Slim II (or TTT-Compact II if you’d like to go a little above budget). It is the finest made turntable I have ever come across, more finely manufactured than turntables that other builders list for significantly higher prices. Built by a very intelligent engineer trained as a Swiss watchmaker. Hard to top that, and the Slim comes in under $10k.
Turntables don’t effect genres...I’ll just say the Slim spins the records at the right pace and uses battery power to ensure as little deviation in power supply and as little vibration as possible. There are great reviews online.
My second choice would be a Brinkmann Bardo or Taurus.