I put several Landsat8 images into ArcMap 10.7. I don't get the same kind of coverage (square area) you see in the first picture from the Earth Explorer site. I instead get picture 2: corners, missing, triangular shapes for some etc.
Though you don't state so in your question, I'm guessing you're using the Analysis Ready Data choice in Earth Explorer? Usually, Landsat isn't clipped like this. In ARD, however, the data is processed and clipped to a predefined grid:
You can refer to this post for an in-depth discussion on why Landsat data looks the way it does. When even a tiny portion of the satellite data clips the grid cell you're interested in, ARD data will appear as you've pictured above.
ARD is great for getting for its consistency, but can be frustrating if your area of interest is near a cell edge or corner, as it will be missing from more of the ARD datasets.
Thankfully, Earth Explorer lets you choose the Browse Overlay rather than the footprint, so you can quickly look and see if the data is sufficient for your needs.
If it's simply too inconvenient, as you appear to be working with quite a large area, you may want to specify non-ARD in Earth Explorer, so that you get the entire Landsat image, rather than a the clipped intersection with a particular tile.
More documentation on ARD: ARD Documentation PDF