What do you use for automated testing of Debian package installations and updates?

The tool you are looking for is piuparts, not debci.

debci is for running a package's test suite under "installed" conditions and hence also checks the installation. But it will not report a red "failure" upon installation failures, it will only report a yellow warning "temporary failure", e.g. when zsh became uninstallable after a BinNMU. See DEP 8 for a description of how such a test suite need to look like.

piuparts in comparison checks package installation, removal and upgrades for success, modified "conffiles" and leftover files. See the piuparts results of all official Debian packages at https://piuparts.debian.org/.

You though may want to run both, debci and piuparts on your packages, if at least one your package has a DEP-8-compliant test suite.