Is the act of the Imam who confines himself to saying "Allah" instead of the takbirat in prayer considered valid, and is prayer behind him valid if he insists on that?

What the Imam does by secretly pronouncing the takbir is contrary to the Sunnah. If he pronounces it silently, then his prayer is valid, as is the prayer of the congregants, because the Imam's audible pronunciation of the takbir is mustahabb (recommended) and not a condition. However, if he merely thinks it to himself without uttering it, this invalidates the takbirat al-ihram, and the Imam must pronounce the takbir in its prescribed manner. The ruling on praying behind him depends on whether his action invalidates the prayer or not. If it invalidates the prayer, then praying behind him is not permissible. If it does not invalidate the prayer, then there is no harm in following him.