Turntable advice

The Rega RP6 is a fine turntable for the money. There are lots of cartridges that will fit it without VTA problems. And in that regard, it's worth reading Roy Gandy's paper on VTA adjustment and the significance) or otherwise, thereof.

In regard to phono stages, while outside your budget at new price, a pre-owned Ear 834p is worth considering - the are extremely reliable and a great buy used at a little over a thousand dollars.

+1 Technics 1200GR!

+1 for VPI Classic

Plan on spending some money on record cleaning equipment! A carbon fiber brush at a minimum. Without it, you won't be able to enjoy the music without the distraction of clicks and pops.

Caution: The VPI Classic recommendation above is not what you want as you are specifically asking for ease of cartridge adjustment. At first, It will take you hours to get it right, more like days. Of course, after 9 years with one, it is a piece of cake and I get it done under an hour. Digital is easy, analog is hard. Just know that before you dive in. Less expensive tables with less adjustments will ease the process but you will regret it later as your options to get a sound to your liking will be limited. You are thinking wisely when you state a table with no VTA or azimuth adjustments turns you off. It should.