We were able to use low amount of frequent flyer miles for round trip from US to London. Our planned itinerary:
Stay in London 2 nights
Fly to Naples and stay on Amalfi Coast, 4 nights in Amalfi
Fly to Stuttgart for Beer Festival (this is a must for us), 2 nights in Stuttgart
Fly to Barcelona, 3 nights in Barcelona
Fly to Porto, 4 nights in Porto
Fly back to London

Any thoughts and comments are appreciated.

It would certainly be too many flights too close together for me! Every time you move, it will take 1/2 day or more. It's not just the length of the flight. It's checking out of the hotel, getting to the airport, the length of time you have to be there ahead of the flight, and the same at the other end. I would definitely choose a more reasonable itinerary. Whatever route you end up with, don't stay in London at the beginning. Continue on to the furthest point and put all your London nights together at the end. That at least eliminates a hotel change.

Have you been to any of these places before -- that is a VERY agressive itinerary IMO.

What you actually have is:

London 1 day (plus a few jet lagged hours)
Amalfi Coast 3 days
Stuttgart / Beer Festival 1 day
Barcelona 2 days
Porto 3 days
Fly back to London (hopefully you are not flying back to London on the day of your flight home - you really need to be in London the night before - which would cut Porto back to 2 days)

Which are your 'must' destinations? You have two weeks and London ( because of the flights) and Stuttgart are set in stone. I'd pick one or at most 2 other destinations max . . . and add some time to London if you've not been before

Was posting the same time as SusanP and I agree -- all your London nights should be lumped together - preferably at the end because you need to be there the night before flying home.

That's one of the most bizarre itineraries I've ever seen, if I'm to be really honest.

I'm sorry if that sounds harsh. But this forum has lots of people who might be able to help revise if you're keen to do that.

Above all, though, it's your trip so don't be too persuaded by what other people think.

we could go to Naples from London same day we arrive. Spend two more days there before Stuttgart. Then go to Barcelona for 5 days and skip Porto. Then back to London for a few days before our London flight home. Basically remove Porto and have more time in Italy and Barcelona.

Your destinations are still far-flung, but that is at least an improvement. At least it cuts out one flight and gives you more days in each place. I assume you have specific things you want to do in each place?

""we could go to Naples from London same day we arrive. Spend two more days there before Stuttgart. Then go to Barcelona for 5 days and skip Porto. Then back to London for a few days before our London flight home. Basically remove Porto and have more time in Italy and Barcelona.""

That makes MUCH more sense. Get the long first travel day out of the way.

But it is still pretty rushed. You have 15 nights from arrival to departure - correct?

5 nights in Naples/Amalfi, 2 nights in Stuttgart, 5 nights in Barcelona -- leaves you only 3 nights/2 days for London. If you don't care to have much time in London than this new plan would work.

What are your exact travel dates? The beer festival is 25 September through 11 October and attracts about 4 million people, so you need to secure tickets and lodging ASAP. Agree with everyone else. Go straight away to the Amalfi Coast before things start to wind down.

If OP buys round trip to London with miles,and then adds the intra-European flights seperately this will raise the question how many hours in London should be calculated to connect from the flight from the US to the service to Naples.
Without Porto on the list, I think the rest is quite doable and nothing out of the norm, especially since flights are still cheap with low cost carriers.
3 nights at any location is a typical weekend getaway, and that's what Barcelona or London are for many people in Europe.
Small airports like Stuttgart are less hassle and less time-consuming than major hubs like Heathrow, CDG, or Frankfurt.
Nevertheless, if OP plans to travel "carry-on only", I would check if it will meet the limits on intra-European flights. Usually you have stricter limits re. weight and size, and those can be enforced very rigorously.

Yes but for most of us in Europe going back and forth is relatively easy. I can fit in a trip to Barcelona at least once a year . The OP is likely not going back to these cities again any time soon.

The other thing is one weekend away is easy. Five in a row ends up being a grind.

I don't find four locations (not five) in 2 weeks that spectacular.
Especially since Stuttgart is just a functional stop to get to the beer festival.
And there is always the problem of how to use a limited number of days.
The initial plan had been very rushed, but the revised plan looks pretty much okay, I would say.

Too many flights. Slow down, rethink, take the train. Skip the Iberian peninsula.
The world is in crisis and short haul flights do nothing to help.
Make the journey as important as the destination, part of the adventure, and use trains where possible please. Europe has a really good network of trains you can use instead of flying all over it, and missing huge chunks. Lower your carbon footprint while you are at it.

London 2x
Naples
Stuttgart
Barcelona
Porto

Five or six depending if you want to count London once or twice

I'm with hetismij2. If London, Stuttgart and Amalfi Coast are highest on the list and you want to add something more, than perhaps choose somewhere between Stuttgart and London, taking the Eurostar back to London. Even that's too many flights for my tastes. Flying is such a hassle in my book and an environmental problem. Europe is chock full of wonderful places to visit and you miss so much if you fly past them.

I thought OP had already erased Porto from the list. See post #7

The itinerary is beginning to look better on paper. On the ground, though, with all those hours to and from and in airports and planes instead of enjoying the sights, sounds, and tastes of new places, you couldn't hand it to me free on a silver platter with VIP service included.

I dislike airports, changing pressures and CO2.

If the beer festival interests you why not go to the world's biggest wine festival in Bad Durkheim? Then you can catch the train between beer and wine. I'm not sure where you are from, but the whole trip looks like you've just grabbed a tour guides list of top 5 things to do and are trying to do them. It lacks concentration or effort.

No Venice/Florence/Berlin/Dublin/Copenhagen.

I'd start the other way around and tell us about your interest, the exact period of travel and what you want to see.

Flygskam.