*: upgrade Go to 1.27.1

## Development Task

Upgrade all active Go version declarations and build/CI toolchain references on
master from Go 1.25 to the latest Go 1.27 patch release, Go 1.27.1.

- [ ] Update the root and nested module `go` directives:
- `go.mod`
- `client/go.mod`
- `tests/integrations/go.mod`
- `tools/go.mod`
- [ ] Update GitHub Actions Go toolchain versions.
- [ ] Update the Docker builder image.
- [ ] Run module, build, lint, and targeted test validation and address any Go
1.27 compatibility issues.

Start with go.mod, client/go.mod, tests/integrations/go.mod, and tools/go.mod, then inspect the GitHub Actions Go versions and Docker builder image. Update every active Go reference to 1.27.1, run the module, build, lint, and targeted test validation, and check for compatibility issues before considering the task done.

Written by the indexing model from the issue text.