Where to go from York/London?

Go to Manchester and/or Liverpool if you want to keep it football-themed.

Ireland would certainly not be an obvious choice to me. You would need to get to an airport and then fly and then the reverse the day after. Only worth it if you can fly home from Dublin and I assume you can’t.
Scotland is equally nor a great plan, travelling 200+ miles in the wrong direction just before having to get back to London.
I would probably plan to stay in York and plan 1 or 2 day trips from there before the train back to London.
You could do a trip to the coast at Scarborough and/or Whitby? Or a trip to Manchester which might interest the kids re the football connections and has lots to interest mums as well. Both easy to arrange and no need to change base again.

Liverpool is another good choice but would add another hour or so each way to the train trip.

How long will you have been in London before you go to York? You won't know this far out for July but frankly with the threat of train strikes I'd plan to be back in London a couple of days ahead of your departure flight. Don't plan it too close so you have some flex time.

Neither Scotland or Ireland make sense logistically. Both York and London have plenty to see. I would explore either of those cities more. I prefer York only because it is smaller and less stressful for me, but I have great times exploring London via the Tube, too. It you want to see a beach town Brighton is a classic and easy to train from London. I would suggest studying both cities & picking out 5 or so options and putting it to a vote of the entire group to see where the most interest lies.

Brighton is the wrong side of London so a lot of effort to get there. And besides Scarborough is much better anyway.. At least it has a sandy beach.
I also wouldn’t be too concerned about rail strikes. The latest two dates, for this Thursday and Saturday, have been called off pending consideration of a new offer on the back of one agreed by other rail workers recently.

As you're already in York I would look at the seaside towns on the Yorkshire coast- Whitby, Scarborough, Filey and Bridlington.
Each has their own character, all have lots of accommodation, and lots to see and do within very easy distance, and all are reachable within an hour or two by transit (I'm assuming you don't have a car) direct from York.
All are long established seaside resorts, so have everything you need.
From either Scarborough or Whitby you can do the North Yorkshire Moors Steam Railway (Whitby to Pickering). Whitby and Scarborough both have boat trips from their respective harbours. All have good coastal walking.
Currently there are no planned rail strikes. After acceptance of the Network Rail offer, and other unions accepting the rail companies' deals the RMT are now the outlying union, so there is quiet hope.
LNER always runs a service from York to London on strike days, and you can easily get from any of these 4 towns to York by bus, so I do not see the threat of strikes as a significant worry to you.

I'm not sure putting it to a corporate vote is really the way forward. With respect I am not sure how familiar with the North Yorkshire Coast most of the American members are, so they will be biased towards the areas they know.
Nor am I sure whether the Yorkshire coast is in RS guides. Just because they are not does not make them worthy of consideration.

You're looking too far away, do NOT try to see as much as possible while you're there that is a recipe for disaster.

Look closer to York and London, flip through Rick's book and find things that sound interesting. England is not really the destination for pretty beaches, and they are all far south of London.

If you're truly unfamiliar with traveling, you need to get researching. And if you decide to take package tours or daytrips, there is no shame in that.

How may days is the soccer group in London? In Bath? Are you flying home from London? If you only have a very few days in London and need to return there from York to fly home eventually, I'd argue for divvying up the extra time in York and London or spending it all in London. Yes, there are fascinating places all over England (some of them mentioned in this thread), but I would suggest they are not going to be more interesting than a 3rd, 4th or 5th day in London, and going to a third city is going to require transportation that cuts into your sightseeing time. There are interesting neighborhoods all over London, some of them feeling very unlike a big city. Just because you stay in a city for more than a few days doesn't mean you're no longer seeing something new.

England is not really the destination for pretty beaches, and they are all far south of London.

Excuse me, that is not being said from a position of knowledge. It's just not true. Tell that to the people who go to the Yorkshire Coast, or many other beaches in Northern England, not to mention coastal Wales.

As you’re in York and you only have 1-2 extra nights before flying out of Heathrow I would not try and go very far. Will you be taken on tours of York and the surrounding area in between soccer activities or is the the trip very sports focussed? You might just want the time to see York properly or, if you will have seen quite a bit then consider going to Durham which is a beautiful small city, or Newcastle which is a much bigger city with a proud football culture.

England is not really the destination for pretty beaches, and they are all far south of London.

Britain is an island. We have beaches everywhere and many are very beautiful. You are not going to get a Med style beach holiday because of the weather but it’s still fun, just different. Bring a fleece, a flask of hot tea and a windbreak and you’ll be all set.

Manchester.

The National Football Musuem is there:

https://www.nationalfootballmuseum.com/

The Mining Museum is very good near Wakefield.
Gorge scrambling at How Stean ,up on edge of the Dales
Sea fishing trip,likely at Whitby.Delending which bit of US you are from the Snozone at Castleford might be of some interest or none what so ever

We have traveling all over the UK in rental cars and didn't find Liverpool that interesting.

Whitby and the Yorkshire Moors was amazing.

Also, Castle Howard is between York and Whitby.

I apologize! I was trying to keep my post short and to the point. I did a terrible job of explaining. We will spend 2-3 days at each stop along the the soccer tour. We will get some sight seeing in as well as soccer. Our coach is actually from York. So I'm sure he'll be showing us all the good stuff there! Our soccer tour finishes up in York, but they do have a train all set up to take us back to London to get to the airport if we'd like. So we can continue our journey out of York or London. Wherever we end up for our extension, we will fly out of that city. We were looking at Dublin because it's actually much cheaper to fly home to US from there. Just not sure 1-2 days will be enough to make it worth traveling there (same for Scotland) And as I said above... not sure what to do there other than the long all day tours (Cliffs of Moher, etc). Thank so much for all of your tips! You guys are so helpful!

I don't think your sons would find Dublin very interesting and with the time it takes to get there and check in and out of the hotel,you would essentially just be seeing the airport.

If you stay in York then your nearest airport would be Leeds/Bradford (LBA), or I suppose, East Midlands (now that Doncaster, Robin Hood has closed).
But I don't think either of those regional airports has anything useful for TA connections, or not that's hugely evident on their websites, except for LBA has Dublin flights (both on Ryanair and Aer Lingus). LBA has frequent trains to Leeds, then a frequent airport express bus from outside the station. Ryanair look to be around half the cost of Aer Lingus.
You could do Manchester, and fly out of there (about 90 minutes by train from York).
But if they are going to put you on a train to London, that might well lean one towards using that train transfer, then having the last couple of days in London.
While you do pre-check at Dublin, as others have said you wouldn't get much useable time there, so it's an option to weigh up and think about., especially if Dublin is a cheaper TA flight.
At Dublin there's quite a few nice seaside resorts and beaches within easy reach of the airport on the northern part of the DART and the northern part of the suburban rail system.
There's another thread somewhere about London with kids.
The obvious would be the Warner brothers Harry Potter place at Watford, if they are into that.

Sorry if I offended any beachniks, I also enjoy a good windswept beach but that is not what most people have in mind when they think of beaches.

And not this certainly: Bring a fleece, a flask of hot tea and a windbreak and you’ll be all set.

If you have time when in York be sure to visit Fountains Abby just outside of York. It was the highlight of our trip.

Juliet