Yeah, I totally get it.
Simplify ignores point selection when 'Convert to Straight Lines' is checked (should be a report, I think), and all segments of the path become straight, even if we select specific points of a path.
So for the partial selection the 'Auto-*Simplify method remains' the fastest one... but it requires us to select these points first (with the star-t and endpoints!), and it likes to add handles, so it remains a weak method.

Pathfinder Add method, as noticed, works great only for closed and non-self-intersecting paths....

So for now it’s either plugins or scripts. I tried to resurrect the script I mentioned using LLMs, and it seems to perform well:
https://www.dropbox.com/scl/fi/quswu73rrbiuskp51njms/Iron-Straight-Points.jsx?rlkey=79044v9k98amewj5k9sovqwf3&dl=1
Please try it and see if it fails at any point or underperforms.

I see what you mean! I think I was a bit unclear, so I'll try to clarify:

I think the problem with using Simplify to solve this problem is threefold.

First, Simplify will sometimes change the shape. It's a bit subtle in the example images I've provided, but even that change can cause problems, particularly for delicate Illustrations or logos with multiple layered shapes that need to align in certain ways. As you mentioned, there are ways of adjusting it using the Advanced Panel, but it still won't retain the same shape exactly.

The second (most important to me) reason is that you can't control which Anchor points are removed with the Simplify tool exactly. Again, yes, there's the Advanced Panel, but that doesn't mean you know which points are going to be removed. What if, for instance, I wanted to simplify one part of a shape, but not another? Such as if I just wanted to simplify the the top part of the shape in the screenshots, but wanted to leave the bottom part as-is. I'd likely have to split the shape into multiple parts, then simplify the part I want, and then merge them back together and hope it doesn't make even more anchor points because of any overlapping edges.

Third, sometime's removing anchor points isn't just about making the shape smoother or flatter. Sometimes, I just want to remove part of a shape, like let's say I wanted to remove the knobby bit at the bottom left of the screenshots. At no point did simplify remove that set of anchor points, because it thinks it's an intended part of the shape. Again, no big deal if it's just this one shape, you can go in and manually remove them with the Delete Anchor Points tool, or you could use a Pathfinder Minus Front to remove it with another shape.

The problem again is that it only works on one shape at a time. Let's say you were working with expanded text that had serifs and you had layered it on top of itself for a cool effect or used an expanded offset path, for instance. But, on one of the letters, you want to remove a serif on the side because it's just a bit too much or it's taking up too much space. You'd either have to go in and remove the serif on each layering of the text or on the expanded outline, or alternatively you'd have to go back and start over with the layering or the expanded offset path after manually removing the anchor points from one instance of the text. By that point, you'd have wasted a ton of time.

To me, it just makes more sense to have a click-and-drag + shortcut key to solve these problems.

You can try to use Object > Path > Simplify for this.
In most cases it’s able to detect unnecessary anchors with its Auto-Simplify mode. If it keeps some, pressing the 'Advanced Panel' button (the 'gear one) in the floating bar would allow you to enable 'Convert to Straight Lines' — with 1° for Corner Pont Angle Threshold it’d purge these rather effectively.
Please try the method and write back if you find it failing for you. Please note this can be recorded into an action, and applied rather swiftly.

There’s a report about sometimes Simplify method isn’t effective (although no files were provided to investigate it): https://illustrator.uservoice.com/forums/601447/suggestions/43363323

Also, I want to mention the great VectorFirstAid plugin by Astute Graphics, that can sweep these off with a button click: https://docs.astutegraphics.com/vectorfirstaid/vectorfirstaid-panel/convenience-operation-buttons

Then, there is an old but free script called 'Iron Straight Points’... I can’t find or remember the source of it. A script solution is definitely possible.

A related request:
A way to delete all unnecessary handles on straight lines
https://illustrator.uservoice.com/forums/333657/suggestions/51075547