I think your plan is a good one.
You might consider this instead of taking day trips from Oxford.
LHR, bus to Oxford
3 nights in Oxford
train to Stratford-Upon-Avon for 2 (or 3) nights
train to Bath for 3 nights
train to Windsor, 2 nights
taxi Windsor to LHR
About Oxford: There's lots to do there, as has already been mentioned.
About Blenheim Palace: There's a city bus that travels north out of Oxford to Blenheim Palace. It stops right outside the gates to Blenheim; you just walk up a gravel driveway/parking lot to pay admission and go in. Time it takes to see the house is about 2 to 3 hours. Your admission price includes the extensive gardens and land. It will take much longer if you choose to walk around and explore those. Blenheim has a cafe for lunch, coffee, tea, or a snack, but in my opinion the food is not great. It's "OK" if you happen to be starving. There's a back terrace overlooking the gardens where you may sit and have your tea or coffee. Blenheim has the largest gift shop of any house or museum I've ever been to in England. You may spend a while browsing. Books, scarves, and an incredible range of things on offer. I bought some gifts there for family members back home.
A better place for lunch would be to exit Blenheim and right outside its gates explore the village of Woodstock. There are several pubs there I've been to whose food I can honestly rave about. I recommend The Back Lane Tavern. Then you can catch the bus back to Oxford from Woodstock.
About Stratford-Upon-Avon: You can take the train from Oxford to SUA. The train station is not far from all the things you'll want to see. Walking distance from the train station to some great hotels, too. Which brings me to spending the night there, which is a really good idea. My husband and I will go to England this fall and spend 3 nights in Stratford. We love walking around there in the evening, after all the tourist "big buses" have left, taking the crowds of tourists back to London. See the Shakespeare sights, enjoy a play at the Royal Shakespeare Theatre, enjoy the good restaurants, take a walk. You could spend an afternoon walking around nearby Warwick, with its many medieval buildings and castle.
From Stratford-Upon-Avon, you could take the train to Bath for 3 nights. Lots to do and see in Bath. The Roman Baths and Bath Abbey are top of the list. I recommend a walking tour and a trip up to see The Royal Crescent. Just down the hill from The Royal Crescent is an outstanding restaurant....The Scallop Shell....reservations needed but it is casual dress.
From Bath, you could take the train to Windsor for your final 2 nights.
Since you mentioned this tour--Go Cotswolds--I will just mention that we have taken this tour a couple of times and it's great.
https://www.gocotswolds.co.uk/product/cotswolds-in-a-day/#tab-description
This tour goes to some classic Cotswold locations: Dover’s Hill, Chipping Campden, Stow-on-the-Wold, Bibury and Bourton-on-the-Water. You will hit Stow-on-the-Wold about lunchtime. I recommend The Queen's Head pub; good food.
You had originally mentioned the possibility of including a day trip to Bath during your trip. New for 2024, Go Cotswolds offers a trip to Bath and Stonehenge which leaves from Moreton-In-Marsh (and Stratford-Upon-Avon).
https://www.gocotswolds.co.uk/product/bath-stonehenge-tour/
(This is in case you do not want to spend nights there, but still want to do it as a day trip.)
Consider staying around in the general area of Oxford/Stratford-Upon-Avon/the Cotswolds/Bath for your vacation, and possibly leave York and Edinburgh for another visit. The things I have outlined here will be easy for you to do; important I think, since it is your first visit.
For more things to do around the Cotswolds, have a look at this excellent short trip report:
https://community.ricksteves.com/travel-forum/england/two-weeks-in-moreton-in-marsh-without-a-car