Okay, I'm at home now (was replying from work earlier), so I can give this my proper attention
I've done a LOT of conversions from TS3, so I think I can help you!
First thing- you said in your first post that you can't open .sims3pack files using TSR Workshop- that's actually not true, you CAN open them with TSR Workshop
When you start the Workshop, click on Create New Project. Select the last option, New Import (the icon is a box) and select Next. Click on the Browse button, and then where it says "File Type", change the drop-down box so it says .sims3pack. Find your file and open it, and when it asks if the .package file is a Sims 4 file say No (obviously). Click the arrows and it'll eventually finish and open the file in the Workshop. I was able to open the sims3pack that you linked above perfectly fine. Here's a screenshot of it:
Like I stated above, generally speaking- the only textures you'll find are the grey maps (in the screenshot above, you'll see it's called a multiplier). The other .package file you posted (with the pillows) has those textures because that's the way the original creator made them; they were designed to display those images OVER the Create-A-Style colors the Sims 3 user picked. They are overlay images, like I also mentioned above. So, the pillow has the Overlay image, but there's also that grey multiplier as well, which would be used to color the pillows. See what I mean?:
See how the screenshot has those images in the Overlay tab? Most Sims 3 objects won't have an overlay- generally, they all have the grey mulitplier/texture map. Or the overlay might be just a tiny part of the overall texture.
Recoloring the converted daybed so that it looks similar to the original one is up to you. You can do so using whatever methods you'd like, but I personally use Photoshop, use the grey multiplier layer as the bottom template, then create new layers over that to create the effect/coloring I want. There are tons of recoloring tutorials out there and you probably already know how to recolor things, so I won't bore you about that. But yeah, when converting from TS3, you usually need to make your own recolors...because it was a bit different than TS2 and TS4 as far as texture maps go (because of Create-A-Style).
There's another way you can find if there was ever a colored texture map, though- if you go to the Mesh tab in TSR Workshop, then click on "material", it'll take you to the Material Editor. Click on Diffuse Map, and sometimes, if you're lucky, you might see a colored image there of what the original item looks like in-game. I haven't found a way to extract that image from TSR Workshop (maybe someone here knows how?), but you can use that image as a guide for your own recolors, or, if you're really a bit lazy, you can take a screenshot and then crop the image and use it for your recolor (it'll be really low-quality, though). Here's a screenshot example of what I mean, using an object from TS3:
Anyway, I already checked your daybed for such a thing and it's not there. So you'll need to use the grey texture, and recolor it your own way, using a 2D photo editor. I hope that helps you and hope it wasn't too much information
Like I said, I've done a lot of conversions, so I can probably do this in my sleep by now, lol. Feel free to ask more questions if you have them!