I'm not a fan of inlining versions that only appear once - it does remove lines from this file, but I find it a lot clearer to have all of the versions in continuous place.

For future maintenance, this also seems like this would be more likely to have a dependency get copied in the same version group without moving that shared version back to the versions section.

I'd love to agree with you, but in practice, this almost never holds true 😛.
Having top level [versions] should be the exception, not the norm.
Let me try to explain my reasoning:

I find it a lot clearer to have all of the versions in continuous place.

I kind of disagree on this, but this is a personal opinion. I can't trust top-level variables like this, as I don't know exactly what they are used for, unless I also check where (and if) the variable is actually used/referenced in the file. So it always end up in extra checks and lookups everytime.
I always prefer simplicity, meaning less indirections.

For future maintenance, this also seems like this would be more likely to have a dependency get copied in the same version group without moving that shared version back to the versions section.

Again, in practice this is almost never what actually happens. We almost always update dependencies with automated tools like renovate, dependabot or whatever. And they take care of this for us. There is also Android Lint rules for this, when multiple versions are referenced for the same dependency. (I also implemented custom lint rules to avoid cases like this)
And most of the multi-artifacts dependencies are now slowly migrating to BoMs making the overall process simpler and less error prone, like Jetpack Compose (except all the other Jetpack library groups for whatever reason 😡)

Anyways, I'm not here to impose anything, but only to suggest improvements I feel worthy, so let me know if I should proceed to the un-inlining of versions in this file 😉

I think the inline version makes [version] category meaningless, and it would ruin the TOML convention.
If you want to remove the one time used version, you should be using this below.
This snippet from a gradle version catalog documentation.
https://docs.gradle.org/current/userguide/version_catalogs.html

This is clearly a downgrade. Also, Gradle recently published guidelines that go against this.
https://docs.gradle.org/current/userguide/best_practices_dependencies.html#single-gav-string

the inline version makes [version] category meaningless, and it would ruin the TOML convention.

This is the whole point. Not using what is not necessary.
Which convention exactly? Extracting version numbers does not serve any purpose here.

I'm not wholly against this change. I think it does make it easier to read the dependencies, however, I also think it needs further investigation (does Android Studio break this format by automatically adding in new versions for single dependencies? Does this format work with our own version update tools/processes?)

@SimonMarquis Given that the rest of this PR is good to merge, would it be possible to split this change into a new PR we can discuss further there?

@dturner Ok, I'll restore the versions variables: 967e7ac

does Android Studio break this format by automatically adding in new versions for single dependencies?

What do you mean exactly? The Android Studio intent to add a missing dependency when it sees a known class? Android Studio has always been in a best-effort.
For instance, I tried to reference these 4 classes:

EmojiCompat
EmojiPickerView
EmojiButton
EmojiEditTextHelper

and here is what Android Studio did:

[versions]
emoji2 = "1.5.0"
emoji2Emojipicker = "1.5.0"
emoji2Views = "1.5.0"
emoji2ViewsHelper = "1.5.0"

[libraries]
androidx-emoji2 = { group = "androidx.emoji2", name = "emoji2", version.ref = "emoji2" }
androidx-emoji2-emojipicker = { group = "androidx.emoji2", name = "emoji2-emojipicker", version.ref = "emoji2Emojipicker" }
androidx-emoji2-views = { group = "androidx.emoji2", name = "emoji2-views", version.ref = "emoji2Views" }
androidx-emoji2-views-helper = { group = "androidx.emoji2", name = "emoji2-views-helper", version.ref = "emoji2ViewsHelper" }

So... I'll let you judge if we should follow what Android Studio does 😝

Does this format work with our own version update tools/processes?

Yes is does, you can see it here: android/platform-samples@4c394aa#diff-697f70cdd88ba88fe77eebda60c7e143f6ad1286bca75017421e93ad84fb87df
It is also smart enough to apply the same version update to multiple entries if they were not using a variable.

I'll re-open a new PR with the inlined versions after this one is merged so that we can continue the discussion 👍