New stuff in that price range is not made well and will not hold up at all.

Invest that money in reupholstering the furniture you have now.

Pick a fabric which is a tight weave, can handle "heavy wear" (fabric store can help with that) and is easy to clean (has some measure of a natural fiber in it, i.e.: cotton, woolâ¦).

To protect if from the messes of small children, allow only water while on the sofa; food and juice must be had elsewhere. Never allow milk near any fabric (if it spills and you don't REALLY get it up, it smells like a dead body), OR, if you want to allow your kids to eat and drink on the new furniture, put a blanket over it, or some area of it, and tell the kids that is their designated spot. And make it a blanket which can easily be washed.

Then, when you have company, simply remove the blanket, and you've got clean furniture.

But $3,000 now won't buy you a very good sofa, let alone entire living room set. Choices now seem to be to overpay for poorly made stuff or overpay for well-made stuff.

Either reupholster what you have, or find some antiques on Craigslist, for cheap, which you like the frames of, and get them reupholstered.

Get only tight-back sofas. They're more comfortable, and they don't lose their shape and therefore last longer. Cushon back sofas, esp. loose ones, lose their shape right away, look hideous, are very uncomfortable because of that, and you spending your life punching them into something reasonably comfortable each time you sit down.

A loose pillow back or tight back is very subjective for whats comfortable to one isnt comfortable for someone else. Their again loose back pillow sofas outsell tight back sofas by a large margin, especially in the medium to high end upholstery.
If you reupholster its most likely you will need to replace the seat and back cushions. High density foam that will last for many years does add to the costs, as well as a fabric that is easy to clean and wears. The exception would be a microfiber. Their are many more choices, colors and textures now for performance fabrics and this could be an option. For a comfortable back pillow that holds its shape and last for many years I have recently seen some foam fabricators mix into a combination of latex foam and memory foam. Both is a very high density foam, that is soft enough to feel like a back pillow. I suggest staying with a hardwood frame, and sinuous (no-sag) springs and high density (HR or Ultracell) cushions, for the best bang for the buck. And look at as many high performance fabrics as possible. Look at easy to remove slipcovers that cover a fabric that's comfortable but maybe doesnt clean as well. As slipcovers can be washed. Also try to negotiate with the dealer for better pricing. Of course the best pricing/deals will be online. Try searching for factory direct custom sofa. Eliminating the middle man can usually save 50%. I should add that you can buy heavily discounted fabrics online and most manufacturers will accept fabrics that you provide.

I am not poor, but I have been poor. I also always liked beautiful things. With the exception of sofas, I ALWAYS have accepted with great pleasure hand-me-downs of very good quality or Craigs list kinds of things--acquaint yourself with the best names in furniture in the last 50 years, and look for those names when you are on Ebay or CL.
Think of Downtown Abbey: Do you think these aristocrats bought new furniture? No--they used their Oriental rugs until you could read through them--so if it is good enough for European aristocrats, it's good enough for me. Never buy junk. And when you buy a good sofa, it can last for decades with just re-upholstery every 20 years or so. Really. I'm sitting in my family room. The only thing bought new (many (about 30) years ago) are two upholstered chairs and a sofa, both Henredon, reupholstered several times. Big Baker chair re-upholstered, from Ebay. Clarence House Chintz for about $13 a yard, from an online discount company. Round Hepplewhite leather topped table thrown out by my mother-in-law because it had a cracked leg. My carpenter fixed it. French armless chair, thrown out by my sister-in-law, reupholstered. Black and gilt coffee table, bought on Craigs List. Oriental rug--big, beautiful--bought on Ebay, cheap. I don't mean my taste need be your taste, but you'll find beautiful things that others don't need. Pretend you are inheriting things from your rich family. Put things together little by little as you acquire them and it will create a very organic, interesting room.
I second what Tibbrix said above: Let the children eat in the kitchen or dining room. Let them learn to respect your things.