I built a dyna-kit ST-70 back in the 70’s. I still have it, but retired it long ago and moved on. Don’t know much about the Eico, but I’m guessing it is a somewhat earlier vintage. If you have kept it up (tubes, sockets caps, driver board, etc.) it still may be viable, but if it is in original condition it will need help. If you like vintage tube equipment, it may be a fun project, but you will likely get more bang for your buck with a modern amp. Same with the Apt.
Your turntable is fine, as long as it is good working order. A new cartridge is in order, any of the major brands have good carts that will work fine with your table. I wouldn’t go much higher than an Ortofon 2M blue, Grado Prestige gold, or an AT VM540ML. The Nag MP110 and the less expensive MoFi carts would be good choices too. Anything more expensive will be overkill for your system, and you will probably not reap any further sonic benefits. The good news is you can try out these carts and rotate them in your system; they will all sound different.
When you are ready to upgrade your turntable and phono pre, you will want to look into MC carts and the more expensive MMs that are out there. Last year a bought a Technics SL1200 GR and experimented with different phono pres and carts. The Technics is similar to what you have, but more refined and will squeeze that extra detail out of more expensive carts (and you will be pleasantly surprised how it will make your less expensive carts sound). I am currently rotating an old Grado, a Mofi Master Tracker, and a Hana SL. I can get it to work with a Denon 103 and an old Share V-15, but those carts are not really a good match for the arm - fun to play with but not recommended. If all you want to do is upgrade your front end, I would stay with your Pioneer for a while and get a new cart.
As far as phono pres - lot’s of good choices out there. On a budget, I would recommend the Schiit Mani. Try it, and return it if it doesn’t float your boat. I finally settled on an Odyssey Suspiro, but a Sutherland KC Vibe I had in my system was a close contender. My next purchase will be a PS for the Suspiro.