Just curious, what is your concern on the 7 year old sitting solo? I believe most airlines allow unaccompanied minors from age 5+ though I'm not positive. I'm sure if you prepped your 7 year old to be on their own (and maybe the 5 year old too) and had a backpack with their snack/water bottle/activities or whatever, they might handle it totally okay. I feel like airplanes are a "safe" situation for a kid to be in a different row. They can't wander off, you're likely able to physically see them (or at least the back of their head) and so you know they're safe. We've also always had the nicest people next to my kids when they've had to sit solo, most people I think are happy to help out with ordering their drink from flight attendant, etc.
I DO 100% get your frustration. My 4 year old has had to sit separately from us on several flights though, and it's been fine. I was flying with her 2 younger siblings and because of the amount of oxygen masks, she hasn't been able to be in our row.
I had a major issue with Alaska Airlines several months ago, they had not only the 4 year old separate but also the 2 year old, and this was after we paid for (and selected) seats that included seat assignments. THey refused to budge (the airplane model had been changed so the seats had gotten scrambled around). On the way to our destination (Hawaii, so a 6 hr flight), the kids were at least nearby us and we were able to work it out (someone was willing to change so we could be by the 2 year old). However, I called Alaska pretty upset and they assured is it wouldn't happen on the way home, they'd fixed the assignments.... and yet when we got to the airport, now our 4 year old was 17 rows in front of us and the little guy (age 2) was on his own a few rows back. My husband and I were also in different rows so we switched around so at least he was in the seat behind one of us. They did NOTHING to help us and it was really stressful. What was worse, there was an extra seat next to the 4 year old and they didn't even want to let us sit in it because it was a more expensive seat and they were wanting to charge us an upgrade fee to sit there (even though no one was booked in it). In the end, a kind flight attendant allowed one of us to sit there til my daughter fell asleep (a red eye, so on take off). The woman seated next to her helped wake her up and get her ready to deplane at the end of the flight.
Moral of the story: Even if you do get to select your seats, don't count on it if you're flying Alaska (we won't fly them again because of the experience). Also to the person saying "You shouldn't expect to go with the cheap ticket and get what you want"... sometimes it isn't the customer's fault, and it's not being cheap. I hope people are kind to kids seated alone regardless and I think most people really are.