Do Remnant Cores drop if you havent killed an Architect?

Good question. I did my first play through on PS4, and despite killing all Architects, and finding most hidden caches I ended up 8/9 cores, so really not sure what happened there. Hopefully someone can answer for us.

I've got the same.  8/9.  Annoying.

In my current playthrough, I've only killed one architect of the four, the Voeld one, but I got the other two data cores on Voeld, so I'm on 4/9. There's an architect I have to kill on Eos so that brings me up to five. I know there are at least two on each of Kadara and Elaaden, and one or two elsewhere. so I should get up to 9 without major issue.
Tricky thing is that you can lock yourself out of doing it if you complete the atmospheric vaults without solving all the puzzles inside. So it's a headache for completionists, a bit like that stupid nomad upgrade task that doesn't update properly.

And EVERYBODY is confusing 'remanant data cores' with 'remnant cores'. Some answers on google are, and some people are. They are not the same thing. A remnant core is a crafting resource that you find in some chests, and I think you generally get some from destroyers, and a few from each architect. You can get them from supply caches.

A remanant DATA core is a quest item, very specific, there are 11 in total, they are in fixed locations, usually ONE in each atmospheric vault (except Havarl), ONE on each architect, a couple in specific other locations. You don't use remnant DATA cores for crafting. If the data cores are for something, and they likely are (even if just a bit of dialogue), no need to spoil that here.

I think there is some confusion, there are a limited number of Remnant Cores for the Task: Remnant Cores which are the ones where you need to clear the vaults and kill the Architects. Now those are basically a one-time deal that if you miss in the vault you are not going to be able to complete the Task.

Now there is another Remnant Core used for crafting and I am pretty sure it can be found in other places and from breaking down materials, it is still a rare resource and harder to find, but you aren't in a lot of trouble if you miss a chest in a vault or don't get it from an Architect. I am not sure if you can find them on vendors or where the best place to look for the crafting material would be, but right now I have about 200 of them on my NG+.

Edit:

I am wondering if you might find them in any of the chests you can find at a Remnant ruin, where it is just a chest guarded by Remnants and it will turn blue after looting the one box.

Unlock the Military Cryo perk that places additional caches in game. You'll have all the cores you need for the Helius Armor. 

Check around Remnant sites. You can find those large caches with a glowing Jardaan symbol above them. They will have cores in them. Sitting on 53 cores currently just from caches I find exploring.

Odd, it's been a while but I remember getting 3 at a time in caches near remnant ruins. They have glyphs on them

Killed 2 architects this evening and got zero cores.