Refresh agent skills during `aspire update`

### Is there an existing issue for this?

- [x] I have searched the existing issues

### Describe the bug

`aspire update` updates an older project's Aspire SDK and package references, but it does not update the Aspire skills previously installed by `aspire agent init`. This can leave a newly updated project with stale agent guidance until the user knows to run `aspire agent init` again.

`aspire update` should invoke or chain to the agent initialization flow after a successful project update so installed Aspire skills are refreshed as part of the same workflow.

### Expected Behavior

After `aspire update` successfully updates a project, the command should run the equivalent of `aspire agent init` (or otherwise refresh the previously installed Aspire skills) so skill files match the current skills bundle without requiring a separate manual command.

The integration should preserve the existing interactive and non-interactive behavior of `aspire update`, including appropriate handling of `--yes`/`--non-interactive` scenarios.

### Steps To Reproduce

1. Create or use an Aspire project on an older Aspire version.
2. Run `aspire agent init` and install the Aspire skills.
3. Upgrade to a newer Aspire CLI/skills bundle.
4. Run `aspire update` in the project.
5. Observe that project package references are updated, but the installed skill files remain unchanged.
6. Run `aspire agent init` separately and observe that the skill files are then updated.

### Exceptions (if any)

N/A

### Aspire doctor output

N/A

### Anything else?

The existing `aspire agent init` installation path is idempotent: it compares file contents and writes skill files only when they differ. Reusing that flow from `aspire update` should avoid unnecessary changes while keeping agent guidance aligned with the updated Aspire project.

Start by tracing the `aspire update` flow and the existing `aspire agent init` installation path. Check how `--yes` and `--non-interactive` are handled, then verify the successful update path refreshes installed skills without unnecessary writes and preserves existing interaction behavior.

Written by the indexing model from the issue text.