Vinyl***What If***

Since you have an MM phono stage, stick initially to a MM cart. I really like the grados, and the "back" or Green" is a steal at about $75-100. I actually have a Green is a very hgih end (albeit legacy) table/arm combo and I'm surprised at how good it can be.

Several companies make integrated turntables. Rega is a good one. I don't know what you call reasonable. Noticed a rec for a direct drive turntable. The poster did not justify why.  belt drives isolate the platter form motor vibrations, but can be more susceptible to slippage and speed variations. Most better 'tables are on fact belt drive and i would have nothing else.
here's the simple fact: setup is as or more important than the hardware. Level.  Aligned.  really aligned.  if you don't like to do such, buy it where they will rally get it right, and make sure the table is perfectly level, and on a solid foundation that wont wobble, vibrate, etc.

oh, one more thing. You want cheap but want to hear the magic of vinyl.  While you are at it try being rich on a small, limited income.
Its a tall order.  Vinyl is a fragile medium. Its why digital is so popular - vinyl is a PITA and most often sounds awful. But it CAN be magical.  Rarely magical and cheap though.

Do i think there are good values out there to get magic or near magic?  Sure - in vintage TTs.  An old AR XA, with the bearing replaced and a universal arm mount machined onto it (there are lots), can be very high quality. Old arms, can be had sorta cheap, but the cheapest are (duh) the fiddly ones.   I have a Syrinx. Awesome. Impossible.  The market speaks.
Another great old table is a Logic DM101. better IMO than a Linn LP12.  But you need a good arm for it.  or, get a bargain basement Linn.  Find a dealer with a used complete setup (TT, arm, cart) for say $1000.  They exist, i bought one as a backup. I have no idea if you feel that is affordable, but for an excellent table, arm and MC cartridge, its pretty cheap.
I hear Schiit plan to introduce a "turnkey" table too, with a uni-pivot arm.  Mike knows his stuff. Company can be a little hard to deal with, but hey.
Now, might you find a freind with such a beast stored in the attic that he/she might loan to you so you can find out for yourself?  Of buy a well-sorted system that you can sell for what you paid if you decide its not worth it.
But, especially if you have a closetful of records, and a decent system, you will find that it is.