Something working so well you don't notice it working

I would say that such a product or process is a well-oiled machine: it's not "squeaking", so people don't even notice it running. From Wiktionary:

Something that operates capably through the effective coordination of many parts.

You could also just use well-oiled as an adjective, and talk about a well-oiled process or well-oiled transition.

It runs like clockwork.

go like clockwork

to proceed, progress, or operate predictably and dependably.

Everything is going like clockwork, so we should be ready to start construction by the end of the month.

The conference went like clockwork from beginning to end.

Farlex Dictionary of Idioms. © 2015 Farlex, Inc

............

go like clockwork

Fig. to progress with regularity and dependability.

The building project is progressing nicely.

Everything is going like clockwork. The elaborate pageant was a great success.

It went like clockwork from start to finish.

McGraw-Hill Dictionary of American Idioms and Phrasal Verbs. © 2002 by The McGraw-Hill Companies, Inc.

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go/run like ˈclockwork (of arrangements, etc.) happen according to plan, without any difficulty or trouble

The sports day went like clockwork, with every race starting and finishing on time.

Farlex Partner Idioms Dictionary © Farlex 2017

This will most likely be useless to you, but there's a verse in Tao Te Ching that illustrates this idea in relation to people:

When the Master governs, the people

are hardly aware that he exists.

Next best is a leader who is loved.

Next, one who is feared.

The worst is one who is despised.

(I would have left this as a comment, but I can't leave comments yet)

This is something I've thought about a lot because of my line of work, my best idea for now is "perfectly integrated". Something along the lines of inconspicuousness and compatability. Remarkable unremarkableness!