Part of the problem is--all of these components--TT, cartridge, phono stage, are very important. Saying that the phono stage isn't important, is like saying that your pre-amp isn't important.
Here's what I have, FWIW, that you could probably buy for 50-60% of your $3K budget: Rega P25 w/Benz Glider that I bought used. C-J EF-1 phono stage that I bought used (I suspect that the EF-1 is the best component of the 3). Again, you could probably buy all 3 now for.......$1500 maybe?
Also, IMO, there's absolutely nothing wrong with owning one component that is "better" than the others. FI, the phono stage--I could see shopping for a used C-J or ARC phono stage & spending $500-$1000 depending on the model. I spent a year (!) of research & shopping to get a great deal on my phono stage.
And don't forget a record cleaner, like a VPI, for maybe $350-$425 used.
Bottom line: There's no one right way to do this. Also, buy everything used, whether you buy a used Rega P3 with a Grado cartridge, or a slightly used P5 w/cartridge. (I'm a Rega fan).
My opinionated 2 cents, good luck!
*And, I'm surprised to hear that the phono stage has relatively small impact on sound.*
