How much do I need to spend to make vinyl sound better than digital?

It is hard to see a good analogue set up for less than, say, $5k new.

This is where we lose the forest for the trees. I’ve been guilty of this too, plenty of times. I played with a $500 Fluance RT-85 and $1K Hagerman Trumpet, and records were still enjoyable. Some vinyl albums simply sound better to me than their digital counterparts, and these still were more enjoyable to me, even on such a modest deck. Though, noise floor doesn’t bother me as much as it might for others. 

A good sounding LP is a good sounding LP. It’s not THAT hard to extract much of the beauty from it. But then, do I nonetheless seek better "quality" and spend money wantonly in its never-ending pursuit and re-definition? Hell yeah. 

Such an emotional response from you two. What if I am right on both counts ? Think about it.

For my somewhat modest system, vinyl beats digital… warmer & fuller even though I have spent more on digital. 

Hana ML > Rega Aria vs Aurender N150 > Qutest with Sbooster Ultra via Qobuz. 
IMHO, while all things matter, your cart is the dominant factor in the sound of your vinyl. Your DAC the dominant factor in the sound of your digital. 

It’s usually the other way around .  A modest say $3-4k turntable setup is usually more natural then a digital setup of the same costs .

when talking reference or upper midfi around $$12k on  up for just dac streamers 

not including LPS for router,good Ethernet hub,quality Ethernet cables , or USB cable which could be another $3-6k more.  Turntables  can sound very good 

but very limited in music assortment ,and takes some manual work to maintain.

this is where digital starts to get much more detailed and refined over the $12k on up mark,.and S/N ratio much better, Dynamic range, better, Low Bass , that’s  whern digital starts to get very good  and just get better but the % of better is at a much $ Higher cost ,diminishing returns.

I much prefer digital to vinyl, but it’s possible to get good vinyl sound but at a price.  The entry level setup the OP describes won’t even get him close, which I suspect that he realizes, and therefore is wondering what the floor that he needs to spend is.  At the bare minimum get a separate phono stage.  I would also get off the entry level cartridge.

  Personally I really like my Technics Direct drive table with my Cambridge Audio mm only phono stage.  A $-300 investment for a good mm cartridge gets much better than entry level cartridges, especially in the treble.  If the OP prefers mc cartridges prepare to spend more.

  These prices are pre tariff.

   I would therefore say spend at least $2K on a vinyl front end or don’t bother.  More money will sound better.

  With digital it’s easier to spend less to get satisfactory results.  $2K can get a streamer/DAC that would leave most people happy for life, although this being an audiophile site, spending more gets into rarefied sound.  With analog $2K gets the minimum-the floor- but leaves you a long way from the ceiling