If your were to assemble a vintage system ?

This is right in the wheelhouse of our little shop here in Austin. We restore pre 1980 amps, receivers, and turntables for clients. And in our mind, "vintage" ends around 1980.  A few suggestions.

First, any vintage amplifier or receiver should have recently replaced electrolytic capacitors. It’s a waste of time otherwise. No doubt the reason that some have had bad experiences with period gear. You won’t hear it as intended otherwise.

Some speakers age better than others. Classic Klipsch usually need little or no attention, many other mid centuries use foam surround and NPE capacitors that need replacing.

Unless vintage cart has a new (not NOS) OEM stylus, it’s best to go with new here. Hard to beat an Ortofon 2M Red or Blue for the money.

Some examples with alternates we think will take on many a more expensive modern one:

Turntable: Technics SL1200II / P2 as alternate
Amplifier: Sansui AU 717 / Luxman R-1050
Speakers: ADS L 710 / Klipsch Heresy (original)

This particular combos might be tricky to keep in the OPs budget, but it would be well worth it.

AR 2ax speakers
Fisher 450t receiver
Sansui TT with a Stanton 681 EEE
cartridge. Put Last on all new records before they were played.
With fresh ears...in my teens,
the best sound I remember. 
Im now listening to Pass, VPI and Monitor Audio. 

Braun speakers (719 or 810) ADS too.

Marantz 2225 with the VU meters

Thornes TT? Something.

All on eBay and with in you range.

BUT!!!!

I’d get new for the down time and future hidden costs that are certainly part of the rig. You’d have to stream. Peachtree makes an intergrated with bluetooth (use yr. phone or laptop). There are others too. Maybe some klipsch speakers new or used. 
Lots of used cables/speakers to snag. With lamp cord vintage it is!
Caveat emptor!

Amazed that no one has mentioned Harmon Kardon. Also amazed how many liked Pioneer. I never heard one that I liked and I heard a lot of them. And I cringe at Luxman. Worst purchase of my life. But we all have different preferences and hearing I suppose. I had/still have a HK 730 twin powered receiver that I believe would shame some modern separates. But It NEEDS to be recapped and some minor repair. So you MUST include that cost in any purchase, which jeopardizes that $1500 limit. Friends, This stuff is 40 + years old. 
That said, my system would consist of

HK Citation  amp & maybe preamp
Infinity Speakers
Technics 1200 TT

Sorry, I don't have model #'s

@artemus_5 

You're right, the HK 730 is an excellent unit.  If you already own one, these are well worth having redone, or buying one.  If you don't, just as with an old car, a buyer would be better off getting one that has already been restored. 

Good separates, HK or otherwise, would likely blow this budget.