Recommended cheaper vintage SS Integrated Amps?

I had that Kenwood as part of my first “serious” system. Nice amp!

   Sansui, JVC, Marantz, and some others are definite prospects. Like stated before, the integrity of components (caps/etc.) is a given. However, I would not purchase a vintage piece just based on the word that the previous owner had all components checked out and, if necessary, replaced. That can be a daunting task unless you personally know the seller.
   Many years ago I had a Pioneer receiver from the SX series which was a decent performer. While it was rated as a "receiver", given it's internal tuner, which was sub par at best, it still produced pretty good sound. All of the previously mentioned brands depended on the speakers they were mated with.
   One avenue you may investigate when it comes to vintage stuff.....Years ago Dynaco offer some products at very reasonable prices. Their equipment was offered as ready built or as kits for those with some degree of soldering skill and ability to read simple instructions. Some of those units sounded noticeably better than more expensive "name brand" products. Don't rule out Dynaco.
   Good luck in your search.
 
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I don’t normally post, but I can speak with experience on this topic, so I’ll jump in.  I’ve had a bunch of vintage pieces, and still have quite a few. Receivers are certainly more popular from that time frame. Based on your request and my use, I would recommend a Harmon Kardon 330c. Not a lot of power, but if your speakers are reasonably efficient, it should play nicely. I’ve compared this unit in stock condition, to quite a few higher powered units, and other brands, and it’s always been way better. Warm but clear, nice deep bass, and visually not as over the top as others from that time frame. They don’t fetch much money, as everyone is looking for Marantz, Pioneer, and Sansui. The 430 is reported to be even better, but I have no hands on experience with it. Good luck on your search. For what it’s worth, with the right speaker pairing and source, the 330c can be very satisfying. I have wondered more that once why I continue to spend more money on gear that doesn’t sound that much better. 

You didn't define the ceiling on "cheaper," but an alternative could be that you get an affordable, modern, neutral amplifier and a tube preamp within your budget.

Many great choices out there including Sansui as mentioned. 
I feel of all my SS restored vintage integrated amps (20 or so) the Kenwood 7100 is the peach without breaking the bank.

As long as a pre/pwr loop is not needed, it seems to make the best sounds with a variety of speakers and overall more reliable than the KA-8100 which suffer from noisy Filter switches (buttons).
The Marantz 1060 is also worth mentioning as its MM phono section is a very good, but overall it is a bit darker overall by comparison.

Not sure just how similar these are to the KA-3500 & 1070 although I suspect similar characteristics with the Kenwood being Direct Coupled & the Marantz capacitor coupled.