If the pheyx pro 5000 is working for you then I wouldn't suggest changing. When I tried pheyx I got too much static. I like Innopow. They have something similar
https://www.amazon.com/innopow-4-Channel-Microphone-Distance150-200Ft-Frequency/dp/B07W5PKR5F/
The Yamaha Stagepass BT600 looks good. I use equivalent Mackie speakers. None of the KJ's around here use towers. The only thing I don't have but I've seen make a difference with the sound is a single powered subwoofer.
The Chauvet Gigbar2 is very nice. I have two. They work best behind the singers with simple static white lights in front to light up their faces.
This is the TV stand I use
https://www.amazon.com/gp/product/B0057PQ8R2/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&th=1
I wouldn't run the audio from the TV into the speaker. I would suggest running it from the MacBook.
I always recommend to use your AV system only for the karaoke music, not to sing through. It was never designed to be used this way. You could damage your equipment, especially your speakers. It's just not worth the risk considering how inexpensive a powered PA speaker costs.
A typical powered PA speaker would be something like this
https://www.amazon.com/Alto-Professional-TS408-Bluetooth-Loudspeaker/dp/B0BB15J4TZ
3 or 4 wireless mic set up
I like this brand although I've never used this particular model
https://www.amazon.com/innopow-4-Channel-Microphone-Distance150-200Ft-Frequency/dp/B07W5PKR5F
song selection/control on iPad or laptop - I recommend using karafun run on a Ipad or laptop. Singers can browse the catalog or select songs using their phones.
video/lyrics display on TV - karafun has the ability to do this, especially using the HDMI output on a laptop.
equipment (mic receiver and mixer??) mounted to existing AV rack
Most mic receivers should easily be mounted inside a rack.
But the mixer is going to be a challenge. I can think of only one way to accomplish this.
A digital mixer with the ability to be controlled remotely like this https://www.amazon.com/Behringer-XR12-Tablet-controlled-Digital-Mixer/dp/B010NR6FS0
An alternative would be using a typical mixer mounted on a sliding rack shelf like this
https://www.amazon.com/Raising-Electronics-Sliding-Server-Mount-Adjustable/dp/B07JN8TG9Q