What would be the effects of installing pitot tubes on the engine covers?
There is a very good reason to install pitot tubes in a place with relatively constant environmental circumstances: calibration.

Image source

When installed at around the nose of the aircraft, the pitot always have a good view of the airflow, at all angles of attack and of sideslip. These parameters do distort the pressure reading, that is why during aircraft development the prototype tows a flying pitot tube and compares measurements. These measurement sets are used for calibration.

If it was installed on the engine covers, not only the aircraft state but also the engine state would have influence on the reading. A nightmare to calibrate, and prone to unforeseen errors for instance if a fan blade breaks off and the wobbling fan causes an airstream situation that was not accounted for.

Best to use the prime location: the nose.