Basic Geometric intuition, context is undergraduate mathematics

The viewpoint you're groping towards is not crazy, and at least historically you're in excellent company -- e.g., Leibniz had similar ideas when he viewed an integral as a sum of the heights of infinitely many lines, weighted by their infinitesimal width, and this intuition is still the background for the notation $\int f(x)\,dx$ for integrals. However, from a modern perspective this viewpoint carries a large risk of accidentally concluding nonsense by stretching it beyond what it can do. Therefore it is not really in favor anymore.

A better answer to your misgivings (in the sense of being closer to the mainstream presentation) is probably simply to jump in with both feet and declare that a line is not really made of points.

Decide to think of a line as something that is in principle a different kind of thing from a bunch of points glued together. You can do this and still acknowledge that points exist and some of them are on the line while others are not.

It then turns out that all of the properties of a line segment (or a smooth curve in general) can be recovered from knowing which points lie on it and which don't. This doesn't necessarily mean that the points make up the line, but merely that the points tell us enough about the line.

It is technically convenient, then, to speak about the set of points on the line as a placeholder for the line itself, when we're formalizing our reasoning -- for the pragmatic reason that we have a well-developed common machinery and notation for speaking of sets of things, which means that we don't need to introduce a new formalism for an entirely different kind of things.

Some people are so comfortable with this representation that they happily declare that the line IS its set of points -- but nobody says you have to think of it that way. As long as you agree that the set of points determine the line, you can still communicate with people who prefer the other idea.

This is less an answer and more of an extended comment. You seem to be struggling with the idea of a point as contrasted with an infinitesimally thickened point, and it sounds to me like you want to do geometry with with infinitesimals. Whereas Omnomnomnom suggests looking at nonstandard analysis, I would suggest a different approach to infinitesimals, namely smooth infinitesimal analysis. It's pretty intuitive (no pun on intuitionistic logic intended) and easy to use. I personally think in terms of synthetic differential geometry and smooth infinitesimal analysis all the time when working with smooth manifolds and Lie groups/algebras. If you're interested, have a look at John L. Bell's A Primer of Infinitesimal Analysis. He's a prominent philosopher of mathematics and I'm sure you'll find something in common with his views.

I don't know if your post has much to do with "life advice", but the question of whether there should be an "infinitely small but non-zero width" is something that bears answering.

The way math is done (with the standard set of axioms), it is indeed taken as fact that a point has exactly zero length and that a line, which has non-zero length, has exactly zero thickness. With an understanding of measure theory, one can see that putting "uncountably many" "measure-zero" things together can produce something with non-zero measure, even if that seems a bit counterintuitive.

However, "non-standard analysis" (which, as its name suggests, is not standard) does allow for a notion of "infinitely small but non-zero" quantities. I don't know enough about this, however, to judge how well the usual formulation of those ideas lines up with your intuition.

As far as life-advice goes, I'd say don't be afraid of not knowing, even if you're the only one in the room who appears to be struggling with the idea. Most of the time, other people in your situation are struggling with the same idea, but these things are hard to express and potentially embarrassing to ask about.

You are certainly not the only one to struggle with the notion of "of course there are infinitely small things!". For example, you'll find that the question of why does $0.999\dots = 1$ has been asked an re-asked many times over on this site.