I second the Rotel RCD 1572 MKII. It’s the single play CD player in my system. It has a good DAC.
I would encourage you to try streaming. I’m over getting up and down too to change the disc. Old guys like us need to try new things that will help.
To start you can get a free trial with Qobuz (actually all services have free trials for about 30 days but I found Qobuz to be the best in my experience). Start with an inexpensive but decent Streamer and DAC combo like the Bluesound Node. You can find a lot of used ones for about $200 US. Easy to setup and use. The Node comes with cables to connect to your integrated amp or preamp. They have a BluOs app you can use on your phone or tablet to choose your music on Qobuz and run the Node. You will need a wired or wireless wi-fi connection. Once you start streaming you’ll never look back based on my experience with friends. So much old and new music available at your fingertips.
Once you decide streaming is for you, you can upgrade your system based on your budget. Now that I’m all in on streaming and ripping my CDs to the same music server, I have a Innuos Zen Mk3 streamer with a 1 TB storage to save the CDs I rip to the unit. Connected to a Holo Audio Cyan 2 DAC. Bought the Innuos used so both cost me about the $3K you have budgeted for a CD player.
Enjoy your new musical journey and please share what you decide to do.