omission of relative pronoun

The color of house that I built is red.

This sentence can be written in following way.

The color of house I built is red.

I can omit 'that', because 'that' is indicating to house, and it is object of the verb 'built'. Relative pronoun can be omitted if it indicates to object. But if it indicates to subject, it can not be omitted. Am I right?

Let see the following sentence.

"Apparel engineers are responsible for establishing and monitoring processes essential to maintain product consistency on time production and fair treatment of workers."

In this sentence, I think 'that' has been omitted before 'essential'. I can write this sentence in the following way.

"Apparel engineers are responsible for establishing and monitoring processes "that are" essential to maintain product consistency on time production and fair treatment of workers."

Is it correct? Here 'that' has been used to indicate to 'processes'. In 'that are essential to maintain............." the relative pronoun 'that' is a subject, not object. Please tell me why 'that' has been omitted before the word 'essential'? Please give proper explanation with some effective examples.

Thanks to all.