Third party consolidators? This is confusing.
itasoftware.com shows various airline prices. It is a service for airlines, owned by Google, but the customer has to make his/her own links to buy anything.
Expedia, Cheapoair, and the like, sell tickets for the airlines. Regular tickets, not some grey-market points-driven tickets. The only "consolidation" they do is for itineraries that require transfers from one airline to another that don't sell each other's tickets.The old bucket-shop days are over. Airlines rarely dump last-minute tickets on a discounter as happened three or four decades ago. Which means the on-line agencies cannot sell cheaper than the airlines themselves. They are not in business to lose money.
On the proposed route for next May, I see tickets on Expedia (my sample) for a few dollars more than listed on American's own site. The American/BA combo, requiring a stop on one trans-Atlantic crossing but not the other, is about $550 US, remarkably cheap in my view. Delta is about $100 more. The only non-stop both ways is on Italia and costs double what American is offering. Expedia's site, in my experience, is awkward. Buying direct from the airline is always safer and more efficient, especially if there is a schedule change, which could be possible if you buy far in advance.
Just be sure to use a genuine multi-search function to put the itinerary together.