Regardless, the maintenance burden remains.
If the code you vendored was well hidden so the distro maintainer didn't notice, perhaps the bad guys would also fail to realize you were using (for instance) libxml2, and not consider your software a target for attack.
The distros try, but one complex problem with a project that holds strong opinions and you may not have a fix.
The gnome keyring secrets being available to any process running under your UID, unless that process ops into a proxy as an example.
Looking at how every browser and busybox is exempted from apparmor is another.
It is not uncommon to punt the responsibility to users.
However, when you install Servo, you just install a single artefact. You don't need to juggle different versions of these different packages to make sure they're all compatible with each other, because the Servo team have already done that and compiled the result as a single static binary.
This creates a lot of flexibility. If the Servo maintainers think they need to make a breaking change somewhere, they can just do that without breaking things for other people. They depend internally on the newer version, but other projects can still continue using the older version, and end-users and distros don't need to worry about how best to package the two incompatible versions and how to make sure that the right ones are installed, because it's all statically built.
And it's like this all the way down. The regex crate is a fairly standard package in the ecosystem for working with regexes, and most people will just depend on it directly if they need that functionality. But again, it's not just a regex library, but a toolkit made up of the parts needed to build a regex library, and if you only need some of those parts (maybe fast substring matching, or a regex parser without the implementation), then those are available. They're all maintained by the same person, but split up in a way that makes the package very flexible for others to take exactly what they need.
In theory, all this is possible with traditional distro packages, but in practice, you almost never actually see this level of modularity because of all the complexity it brings. With Rust, an application can easily lock its dependencies, and only upgrade on its own time when needed (or when security updates are needed). But with the traditional model, the developers of an application can't really rely on the exact versions of dependencies being installed - instead, they need to trust that the distro maintainers have put together compatible versions of everything, and that the result works. And when something goes wrong, the developers also need to figure out which versions exactly were involved, and whether the problem exists only with a certain combination of dependencies, or is a general application problem.
All this means that it's unlikely that Servo would exist in its current form if it were packaged and distributed under the traditional package manager system, because that would create so much more work for everyone involved.
So it's ultimately a trade off rather than a strictly superior solution.
Also, nothing in Rust prevents you from doing the same thing. In fact, I would argue that Cargo makes this process easier.