Can relative pronoun be omitted in the following sentence?

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 the house, and it is the object of the verb built. A relative pronoun can be omitted if it indicates the object. But if it indicates the subject, it can not be omitted. Am I right?

Let's 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.