No, it's not always the parents' fault.
And people can only do their best, we're all hampered by our own foibles and inadequacies. I loved my older boys dearly but made some major mistakes with them. Now they're saddled with the backlash of those mistakes..their own personalities, that were molded by them, but the kids I'm raising now are benefiting from them (and never fear, it doesn't escape the notice of the older boys) because I won't make those mistakes again. I'm sure I'm making other mistakes though. To tell the truth, I'm sure it would have been better for the younger kids if I hadn't brought their brother here. But these are the trade offs we make. Sometimes they work out, sometimes they don't. In this case, they get to know their older brother. So we have to walk sometimes and reaaallly stretch the menu (we're eating some interesting meals right now in this last week, I must say).
I was going to read through the thread, then respond to a couple of points. You've brought up one here though, that is important.
Bringing this older boy to your home, may be a good thing for both the younger ones and him. You seem to have the skills and experience to carry it off.
First thing, your op on 'chores', I think it was yours, is a step in the right direction. As both a parent of twenty something kids and more than a decade of teaching experience, kids need chores. In school one can spot the kids who do not contribute at home. They are disorganized, haven't a clue to how to work independently or in groups, and lack self-esteem. While I've only taught in middle and high schools, it's true in lower grades too.
Chores though need to be age appropriate, beginning around 2. It's not easy and the younger they are, the quicker you need to adapt their chores, getting a bit more detailed as they gain mastery. Toddlers can put their toys in a basket. As they get the hang of that, give them sets of baskets: big toys (stuffed animals, musical stuff, mostly 1 big piece stuff that's pretty indestructible), one for books, one for puzzles, and small ones for duplos, etc.
By later 3 or early 4, they can help with picking out clothes and laying them out for the next day. When feasible, they should be putting on their socks, underwear, pull up pants, some can even put on t-shirt or sweater if laid out correctly. They should know where dirty clothes go and put them there. They should put their shoes in closet or wherever you say they belong at the end of the day, (Your kid will not have 2 different shoes because he could only find one left one. They should be able to brush their teeth and have a place to put it away. They can tear lettuce for dinner and set the table. They can help pour liquids into a batter. With supervision they can even stir the batter.
By late 4 they should be dressing themselves in the morning, brushing their teeth, and be ready for the day before breakfast. They should know to wipe out the sink and dry off the counter. If they have an alarm clock that you set the time on and show them how to set the alarm, they can do that. If in preschool, they should know what they need in their book bag and have it ready to go by the door the night before. (Right there your child is light years ahead of the kindergarten, 1st, 2nd, and 3rd grade peers).
By 5 most kids can handle a not too sharp knife or better yet, a vegetable cutter to help prepare salads. They can set the table after you show them, pour beverages, etc. They tend to eat what they help make and especially if they helped pick out the 'dinner food' at the grocery.
As kids get older, though many can make a bed by 5, add to what they can do. From dusting furniture, to vacuuming, to raking leaves, shoveling snow, taking out the trash. Expect them to mark a calendar of what their chores are, what activities they need to be at and when. They should check off when completed.
These types of routine segue easily into expectations in school by 3rd or 4th grade. Kids should be writing down their assignments, including anything they may need to make plans for, (i.e., buy posterboard, need new markers, get to library, ask mom for ride on Tuesday-project too big for bus, need $10 for field trip-in 2 weeks. How many kids ask for that the morning of? Parents don't have it, that's a crisis.)
All the activities tend to make homework a lower priority than it should be. Until middle school though, most activities tend not to immediately follow school, but start around 4-4:30. One of the best and hardest things I did when mine were small was to pick them up from school, tell them to change their clothes, gave them a snack and then homework. They really wanted to chill, truth was though they were still in school mode and wanted to either get out and play or get to soccer or whatever.
Now to my real point in this post. The age of the 'new son' is considerably older than the younger. He'll be capable of much more and that should be pointed out at a 'family meeting.' Think of it as one of those times, "Oh my! I never noticed how strong you've gotten, you are really becoming a young man!" Then to the little ones, "In a few years you too will be able to do what ____ is doing. He can be the one to help you carry in the groceries, while one of the younger ones carry in the gallon of milk or laundry detergent. The younger ones can grab the trash from the bedrooms, bathrooms, while he can bring the kitchen garbage to the cans and the cans to the curb.
Along with more responsibilities come privileges. I believe you mentioned pulling electronics by 11pm? In middle school that's late, really. Homework should be first done. If tv is on the menu, pick together-you having final say and limiting to an hour. Video games should be 1/2 hour or borrowed from whatever tv time you allow. Computer time should also be limited and always where you can see it. 5-7 year olds should be ready for bed and settled down by 8:30. 8-10 year olds by 9. 11-14 by 10.
High schoolers? Well that's tough, but parents should control electronics, including cell phones, ipads, etc., from 10-11 pm on. You can't make them go to sleep, but without ability to converse with peers, they will probably be out by 10:30. A word though, if your high schooler is into extra-curricular activities and hopefully they are; and taking honors/AP or having trouble in school, they may be doing homework until midnight or later some nights. They may need the computer and they need an adult presence as much as possible that late. (Again, the voice of experience. My youngest had the year from hell in his jr year. 4 AP courses, X-Country on an All-State team and he was All-State runner, track & field. Most nights he was up until 1:30 and left for training at 5:45 am, how he didn't come down with mono?) I probably stayed in kitchen 2 out of 5 nights, I had to leave for work myself at 6 am and I didn't do well with less than 6 hours of sleep.
Here's the thing with your 13 year old. Build him up in front of the younger boys. Make sure HE knows he's a model for them. He is the one they'll look to for eating, studying, friends, etc. Most kids, but especially boys, literally are thirsting for admiration: looks, smarts, strength, trustworthiness, kindness...
Most younger boys follow older ones, especially if their talents align. Might be sports, demeanor, manners, etc.
Good luck!