Versus what? Everything you install involves trust at some point.
[1] Standard “GNU”/linux desktops
In both cases, you trust the publisher and in both cases the publisher gets equal access to your machine.
Oh - you mean you're downloading the source code, then audit it, then compile it and only then you run it?
That's super great. That has saved you from the xz backdoor and all other supply chain attacks and will be of great help to you in the future. Let's hope no backdoor ever slips past your code review.
The difference is that the attack vector of the shell script is an easier target.
If someone was to be malicious; they could manipulate the script and inject some sort of payload in disguise. It's an easier vector to damage than say an compiled package. One that's less prone to being detected in that the script could go for days undetected.
With the executable you can compare the checksum and with the whole package compiled it is less prone and more tricky to alter.
Unless that script is under monitoring 24/7, I'm going for binary but they don't support BSD anyway.
It's much, much easier to hide a malicious payload in a binary than an easily auditable shell-script. And it's much easier to make a decision of whether the payload should be enabled or not if you are already running on the local machine.
If you don't trust a publisher, you really can't run anything of theirs. Shell script or, especially, binary.
Even if it's auditable, how many people are actually verifying the shell script before hand?
You've just been given a command to download and execute.
And the potential of having lots of users downloading a shell script has a quicker attack path than users downloading the package. You have custom repos, holding their own distro packages for the software.
In all cases, signatures and repositories need to be configured, often requiring both root access and usage of the CLI and in all cases much harder than running an installer script (which might be doing exactly these steps).
To achieve easy means of installing using distro package managers means including the application in the distro itself, but now it's beholden to the distro's software update policies and thus stuck on that specific version for years or even decades.
That is not what a v0.something of an end-user centric desktop application wants for themselves.
The response of "it depends, we probably used your system package manager" was not often well received. Users who know how to use their package manager tended to just do that anyway, and not use the script.
Also, cleaner uninstalls. If the software only has access to specific directories, I can be reasonably optimistic that removal will be clean.
Furthermore, it is much more convenient. E.g., I can just winget install vscode instead of having to google download links.
Can Linux have something like the Mac App Store where apps don't have access to the whole system by default?
Sadly code editors aren't really suitable for flatpaks, since they usually require access to dependencies installed on the host. This can be worked around by using dev containers, vor the IDE has to ne developed with sandboxing in Kind (like GNOME Builder).