That fixture was never legal.
It might be a foreign knock-off, but more likely it was thrown together out of parts, and some parts are missing/omitted.
Look at the blistering of the drywall paper above the incandescent bulb. It's illegal to sell fixtures that do that to a ceiling. Any fixture capable of taking incandescent bulbs is required to provide thermal protection to ceiling surfaces for the worst-case bulb someone might put into it. They allow a sticker that says "60W max" or some such. I've seen fixtures of a similar style, but they put a lot more space between bulb and ceiling. Perhaps someone "hacked" this fixture to get a low profile out of it.
Separate from that, this fixture was designed exclusively for incandescent bulbs, without the slightest thought for CFLs and LEDs, which have a wide base to house their control electronics. Or possibly, the way the lamp was misassembled caused an unusual spacing of the sockets.
Into the trash it goes.
Start with the color temperature you want.
On the left is 2700K if old incandescent, or 3000K if halogen (bulb within bulb). On the right is 4000K or 5000K - the markings on the LED will say for sure. People think they want colder (higher value) light than they really want. 4000K or 5000K is good workspace lighting, 2700K feels homey.
You can get any color temperature you want, as long as it's not black.
CRI is the quality of the light. Perfect is 100. Incandescent is 100. LEDs are 30-98 depending on what you pay for.
Now you can shop for a fixture you like. You can get sealed "bulbless" LED fixtures, which if competently made (i.e. not cheapo) will last long enough to make sense (i.e. longer than you'll live there). These fixtures have more design flexibility, since they don't have to account for a "worst case" thermal load.
Or you can get fixtures with special sockets like GU10 that will only accept CFL and LED screw-ins, and those too take advantage of not having to worry about a 60W/bulb thermal load.
Or (local Codes allowing) you can get traditional fixtures with Edison sockets, which are thermally rated for all that, and put any kind of bulb you please in them.
Tell the landlord the model you like and say "that one" and offer to help procure it (not the same as pay for it). It sits in a box until the next time the landlord's handyman or electrician is onsite for other work, then done. Pretty good deal for the landlord.