I'm not really satisfied by the answers "they walk around because they're required to do so".
Besides the fact that others might have a lower stake in the health of the plane, it's worth noting that over the decades, the culture of the aircraft industry has been forcefully changed, due to crashes happening for dumb reasons.
In an episode ("Check Yourself") of Hidden Brain, there was a particular focus on a crash at a flight show back in the 1940s or 1950s, where they were introducing the B-52 bomber (in service since 1952). Since it was at a flight show, obviously these were top pilots who were doing the demonstration, but anyway, they pitched up too steeply from takeoff and fell out of the sky.
Some people survived the ensuing fire thankfully. It turned out that some sort of flap was locked in place, which was meant to act as a brake on the ground; and had the effect of increasing the pitch. This would typically be unlocked from the outside, but in this new model, it was done on the inside. It seemed to be that unlocking it skipped the mind of everyone and if they had realised in time upon takeoff, they might not have been able to lean over and turn it off anyway.
The naïve solution of "just be more careful" wasn't feasible − these were accomplished pilots; and pilot training could only take so long. I encourage you to listen to the episode, but to summarise, the main thing to come of this investigation was the decision to start using checklists of 6-10 checks − the top few dumb reasons that might cause crashes.
Over the years there were other scandals too, eg Korean Air, which was largely blamed as a cultural issue. Korea has a culture built heavily around respect for different classes − your boss, your elders. This culture is seemingly more extreme than what exists amongst its East-Asian neighbours; and it's built into the language as different forms of address. If they're asking for something or trying to make a point, they beat around the bush because they're terrified of being too blunt.
As the remedy for all the crashes, the pilots were encouraged into the notion that it's OK to speak up if you think something's wrong; and more drastically, they were also told that from now on, they had to speak English, like all the other pilots and airports were typically doing.
English doesn't have the different forms of "you" like so many other languages do; and similarly, trying to express all this indirect desire in English is avoided because it sounds weird − there's an example of a Korean pilot repeatedly expressing that basically he was a fan of weather reports, in regards to his captain blaming the weather reports for the fact that he was unexpectedly flying through clouds, in the wrong place, about to crash into a mountain.
In another industry, it might indeed seem normal to allocate someone else to walk around the plane. If someone else said afterwards though, that they too were going to inspect it, it would probably be perceived as a sleight − "I already checked it. Are you saying that I'm blind?"
If it were as simple as saying "the pilot has the most at stake, so she has to do it", then we could point to seatbelt laws − in eg NSW, Australia, it's the driver who gets punished, if some of the passengers aren't wearing seatbelts.
In this aircraft scenario, the pilot has internalised the fact that he has the most responsibility. He wants to check the aircraft, and it's not going to offend anyone else who was already expected to notice anything.