What's the weak link in my system?

If you don't know ... then nothing is weak.    Once you hear something, you will know.

There are so many things to consider. Always make your front end the best u can possibly afford. If it starts out crap, u can't hope for anything more at the end. Not saying your cart sucks, but invest in the very best u can. I had the preamp u are using and it sounded good for what it is. High current amps and good speakers are a must. I realize, this comment doesn't give u a lot to go on, just saying. As with women, the front and back end are a must. 

Agree with Handyman.  The front end is probably your weakest link - both analog and digital.  I owned the Bellari and found it lifeless. Did not keep it. Get a good solid state phono preamp, like a Dynavector DV-75, used. I prefer tubes, but good tube phono preamps are expensive. For digital get the best dac you can afford. Everything else should be ok.

Dave,
Your argument is valid if more expensive sources made a difference, but not if they don’t. I am not quite sure where the minimum is exactly, and it may be wise to pay a bit extra to be on the safe side, but sonically perfect sources do not have to cost an arm and a leg anymore. See here for the specs of my usb DAC: https://www.jdslabs.com/products/39/odac-objectivedac/ Those are well beyond the hearing acuity of any human. That is the great benefit of technical progress. Perfection can be defined in relation to that human hearing acuity. There is no possible audible advance beyond that perfection, but such perfection is getting cheaper, and we are there. I admit to a kind of naughty pleasure in referring to the ultracheap Chromecast Audio, but it is indeed the best example of the progress we made, even if it does not measure perfectly (but almost). The good news is that at least for this part of the chain perfect audio is now within reach of all social classes, and not just to those who want to brag about their wealth. So to continue the bragging, they have to invent their alternative facts. I for one will not waste money on the advice of some spoonbenders, against the advice of the designer of my speakers, and against the advice of the designer of the original cd format. Did you actually read the Chromecast Audio test, or any of the other tests on that site? Is anything wrong with the test?
As for the plasma screen, my Quad speakers are dipoles so do not radiate sound to their sides. How else can you watch opera or ballet (or don’t you?). My experiences with live acoustic music do matter (and yes, I did sit in e.g. the Mariinski in St Petersburg) because they provide a yardstick.
There is one thing I forgot to mention about my system, and that is the house itself. It is a detached home in a quiet leafy suburb, and that is important, because detail resolution is obviously greatly impaired by background noise. So when it was designed we set about making sure it was as quiet as possible. Construction is concrete (also for the roof) and brick, and the heating and water piping system were designed to be as quiet as possible (no air heating or air conditioning, which we do not need/use here anyway). The same applies to our modern ultra quiet kitchen appliences. Ventilation openings to the outside are muffled, and the window glazing is specially sound deadening. These are cheap things to do when you build a house, but obviously hard to achieve afterwards. But they make a real sonic difference, comparable to listening in the middle of the night. Money much better spent than on cables or other quack stuff.

I’m an audiophoole on a tight budget. After experimenting with speaker placement and carpet/furniture alternatives, here’s how I’d approach it; just my 2 cents.

I’d focus on vinyl first. A phono pre upgrade would not have to be very expensive. A Schiit Mani or Emotiva XPS-1. The XPS-1 is an especially good bargain and will let you play with cartridge loading. Experiment with different platter mat materials. Cork/rubber, acrylic, none, Teac Washi. Experiment with VTA. This is important: Phono cables should be designed to be phono cables! Any old RCA IC will not do. Usually 22-24 awg silver coated copper with the lowest capacitance possible. They don’t have to cost an arm and a leg. https://store.wireworldcable.com/collections/interconnects/products/luna-7-audio-interconnect-cable?... All this is intended to help you not give up on your TT unless absolutely necessary, because as pointed out, that will run into real money. Try a cartridge upgrade as a last resort; but before you spend that kind of dough, make sure it is also a good match for your dream TT/arm. None of the above would be wasted if you have to bite the bullet and get a new table.

Try to change one thing at a time. The experience may lead you to a common solution for CDs. I wish you well!

Jim