Renting a car outside London to tour the Cotswolds for 5 days in June 2019

Yes that makes sense.

I think renting a car at Gatwick makes sense.

Here are a few factors to consider re renting a car in England:
1. Gatwick rental car vendors are likely to be open for longer hours/more days of the week than rental car vendors in cities/towns.
2. If you plan to rent an automatic, there are likely to be more on hand at Gatwick than at rental car vendors in cities/towns.
3. Reduced hassle factor. It will be so convenient to return your rental car at Gatwick. If you rent the car instead in Oxford (for example), you'll likely have to take a train to Gatwick.

We rented our car from Europcar at Gatwick and it worked out for us. We had a Renault automatic. We stayed in Tunbridge Wells the night before we flew out; it is an easy drive. We filled the tank ( think that was part of the agreement) and washed the car before we returned it. And took a series of pictures of it when we turned it in so there would be no surprise charges.

I don't rent cars in the UK, so maybe there are fine points that I don't know.

I'm confused. When you are going northwest from London towards the Cotswold Hills, why would you go halfway from London to the south coast to get a car to then drive all that way back and all the way around the M25 - thrilling at best? It doesn't make any sense to me.

Why not rent it where you need it? If you can't get a car in Oxford or other locality which would allow return at Gatwick, surely you could get one at Heathrow (an hour or more closer to your destination) which would.

and washed the car before we returned it.

That's very decent of you. Filling the car up with fuel is usually part of the agreement but this is the first time I've heard of anyone washing it!

As Nigel has said, Gatwick is in the wrong direction if you are heading to the Cotswolds, so I would hire from Oxford and return it to Gatwick. Might as well take a train heading in the right direction than get stuck on the M25 in traffic getting round from Gatwick.

Not difficult to pick up a single to Oxford in the region of £5-£6 on the train from London. Most of the major car hire companies are strung out along the Botley Road and would pick you up (at least one is in easy walking distance too). They won't be open Sunday and close early Saturday though.

Another vote for renting at Heathrow or Oxford.