Does an (almost) full passport increase the time you spend at immigration?

In my experience a (near) full passport will normally increase the time you spend at immigration.

How long it adds will depend on the immigration official. Most of the time it will add about 5-10 seconds as they spend time flicking through looking for a page with some free space in order to put a stamp on.

Occasionally they will spend some time looking at the stamps and where you have been, but in my experience this is rarely more than a cursory glance, and might add an additional 15-20 seconds.

Given the non-trivial cost of passports I can't see it being a good idea to renew a passport before it is (nearly) full - unless you consider those few seconds to be worth several hundred dollars. Even then, a near-empty passport can also cause delays - I once had an immigration official flip through most every page on my empty passport looking at (and commenting on) the animals on the pages (Australian passports have a different native animal on each pair of pages. Once the passport has stamps in it, they are not particularly visible, but on a near empty passport...)

I've never had a problem with a full passport. I've even had extension papers added to a passport that was nearly full but had plenty of time left on it. Occasionally, yes, it adds some time (for the pedantic officers out there), but in the vast majority of cases it does not. They simply look at the stamp appropriate for them and move on.

Immigration officers are primarily in the job of inspecting the traveler, not the passport. Watch any documentary about customs/immigration, and this is a recurring theme. When they take their time riffling through the pages, they are sizing you up more than they are really taking any interest in the stamps. If you give them a blank passport, they will often answer it with an equally blank stare that goes on for 30 seconds - again, sizing you up.

It's not that the stamps are of no interest, and at certain ports (e.g. Israel) the presence of certain stamps in your passport can become a real problem. But in general, at most ports the stamp review is a secondary task to their primary task of evaluating the risk posed by the traveler.

So I would consider the question from another angle: will renewing to get an empty passport guarantee reduction of my time in immigration in any way?

No.

The immigrations officer will still take time to consider you. And the length of that time is entirely at his/her discretion.

Moreover, there could still be long lines at customs. Or an immigrations officer could still decide you look a little too much like someone on their watch list and pull you aside for more intense questioning. Or your electric toothbrush could have started itself up inside your checked bag, leading to a tense and lengthy discussion with a baggage officer before you finally convince them to open the bag and let you move on. The passport review is a tiny part of the time cost of customs.

For example, JFK Terminal 4 has a daily peak average wait time of 83 minutes. Even if you reduce your time at the booth from 3 minutes to 1 minute, you are effectively trying to optimize to save ~1% of the time you will be there from entry to the customs hall to final exit.

Not worth it. Just travel normally and renew your passport on schedule. Customs is just that: customary practices. Embrace the local custom and enjoy the journey - a maxim that applies equally well to ports of arrival and to travel in general.