Critical oppositions made using inadequate equipment.

OP: isn't this:

do I need to wait until I get better speakers to hear the differences of cartridges?

Loudspeakers in an infinite variety of rooms are so varied that evaluating other equipment only has merit in a particular system. There are exceptions, a cartridge that does not track well will break up prematurely and that will apply to all systems. Otherwise clearthinker is right, you can not generalize to other systems. Now if there was a calibration standard that could be applied to audio systems like there is for projectors this could change. This is actually now possible given digital control but would not apply to "analog only" individuals. 
In short, we are very much all alone by ourselves when it comes to evaluating equipment. What sounds great in your system may not be great in others regardless of quality. This is one reason you have to take reviews with a grain of salt. You may also notice that certain brands are always being reviewed by people who are fond of that brand to begin with. Bias has many forms. 
How do you keep from buying equipment that will not work well in your system? In the end you can't. I would bet that many if not most of us have bought equipment we became unhappy with. Our systems evolve slowly over time and we learn what works well with our own situation. Set amps will be worse than terrible with my speakers. I know not to go there. With horn speakers my amps are extreme overkill. There are so many ways to skin this cat. We all have our own ideas of what is best. I do find it interesting that if you have an excellent system evaluated by a number of audiophiles all will agree that it is a fine system. The same will happen with a terrible system. All will agree it is terrible. So there is a standard in our heads even if you can't define it.  

@surfmuz if you asked regarding your own system, don’t hesitate to tell us what you have and what you are willing to upgrade

No, it was about somebody else’s system.  I don’t use monitor speakers in any of my home three systems. Next coming upgrade I’m planning is NORDOST HEIMDALL power cable for my Modwright LS100 pream for my basement system. 

Yes, you can. I do it all the time.

I have 107 cartridges, at least that many styli that I evaluate, compare, review, and enjoy. I have 17 TTs that I mount them in with varying arm masses and configurations.

My equipment is not expensive, and hardly high end by most standards in here. 
But I can clearly hear all the subtle, nuanced, drastic differences in cartridges and styli. For set up of cartridges and styli, I listen through a $6 pair of MonoPrice headphones. They readily reveal VTF issues, alignment errors, tracking issues, distortions, and any other problems. I also make styli for several rare cartridges. 
So, yes you absolutely CAN hear differences using inexpensive (consumer grade) TTs, receivers, amps, headphones, phono preamps, cables, interconnects.  I do it and enjoy it every day.