3 way joint out of cylinders

The problem you are showing there is what’s known as the tiny face issue.
Sketchup struggles with very small tolerances, it wasn’t originally designed as software for 3d printing small things, it was for architectural stuff, so very small or very very large can cause problems.
What happens with very small things is that the vertices get so close together that sketchup things that are the same and fails to form the face between them. The tiny faces can exist but it can be difficult to create them.
So there have been a couple of ways of dealing with this depending on your workflow.

Originally people scaled up, edited the shape then scaled down again. This works but is a bit cumbersome, and you can end up ‘walking’ your model a long way from the origin.

Another option is to use a second instance of your component, scaled up and edited, the edits will be reflected in the small version, then the large one can be deleted. This is useful when you want to work on something small in the context of a larger model. As an example, you could make a copy of a door handle, move it outside your house and scale it larger than the house, detail it, the delete it and the original small one on the door inside the house will have the details without ever moving anywhere. This has colloquially become known as The Dave Method although he didn’t specifically invent it.

But realistically if you are designing for 3d printing there is no need for any of these geometry gymnastics. The .stl file format that you will export to print is unitless. Meaning it only knows it is a certain number of units large, those units can be anything. So, you can model in metres and export the stl then tell the printer it is millimetres. This can take a bit of fiddling with your export to understand what settings to export to suit your printer software, but that can be done with a simple cube of a specific size to understand the settings.

So having rambled on with all that, your model is failing because it is too small, work at a larger scale and the faces will form.

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