My CD-player stopped working. Does it make sense to invest in a new CD-transport?

Unless I missed it, you didn't say what player you have and how much it would cost to fix. If it's a good company I would fix it. 

For the price of a CD player, I bought a Mac Mini, a internal disk drive, and a case for it. Now, I have a CD player, a streamer, a computer (obviously), and a media server. And there is so much more I could do with it. $500 for a flimsy little transport. Nope. That was never going to happen

While most of my listening these days is via a Wiim Ultra which I truly enjoy, I have enough cd's to warrant a decent player. I recently purchased an excellent Emotiva player (used) on the cheap and am happy with the choice. I do notice a slight improvement in resolution, I mean slight. But with 1000+ cd's at the ready I felt motivated to keep that tradition going....

I bought a Cd music storage system, a Baetis, with a built in ripper. I basically play my CD's once, and then put them into play lists ( the songs that really work for me ). I have a Main List of tracks, currently over 12,000 tracks, that I hit Random on now and then. It changes up the song order. If I need to "listen" to a CD, I have a Sony Blu-Ray player that will do. My 2nd stereo system in the garage has a 5 disc CD changer, so there is a back up. My Rega turntable hasn't been used in around 12 months.

I prefer to buy used CD players with digital outputs and use them as transports. I am currently using a decent circa 2000 Rotel CD player through my DAC. Solid build and good quality transport. It's onboard DAC sounds a little crude by today's standards.