Fourth-starting Sainz was on course for third place when his Scuderia stable-mate Charles Leclerc was forced to retire on lap 39 out of 57 with a suspected engine failure.
However, the 2022 British GP victor was then bumped off the podium in the dying stages by compatriot Fernando Alonso, the two-time world champion making his Aston Martin debut.
Sainz called out poor tyre degradation aboard his SF-23 as fundamental to Ferrari's inability to mount a challenge to the Red Bulls that bagged a 1-2, with defending champion Max Verstappen leading Sergio Perez.
But he was equally alarmed by the pace of the AMR23, which he called "very concerning", as the injured Lance Stroll added to the Silverstone squad's points haul in sixth.
The Spaniard continued: "I wish that as soon as we go to other tracks where we cook less the rear tyres, we can hold on better.
"It's clear their car has something, both Red Bull and Aston, where they degrade a lot less.
"If you look at Mercedes and us, we have very similar degradation. These other two cars, for some reason, they don't degrade.
"It's something we will have to look into, analyse and see what we can do."
Alonso managed to steal the position off Sainz by finally activating DRS on the run to Turn 11 to settle what had been a fraught battle, with the duo having made brief contact.
On the dice, Sainz said: "It was a nice battle. It nearly, unfortunately, cost me a position to Lewis Hamilton, fifth] also because in our car, as soon as you push a bit to defend from Fernando, you cook the tyres.
"It's problem that we have too much degradation, the tyres get too hot when we start pushing. It means we don't have a lot of margin in the race.
Asked about the coming together with Alonso, Sainz said: "I felt it from behind but to be honest, I think it was a nice clean battle. It was just a little touch.
"Always good fun but in the end, the time put us where we are right now that is behind Aston in the race and clearly behind Red Bull."
