The USGS EarthExplorer tool uses World Imagery from Esri, according to the tile source shown in the image below:
There is also another World Imagery tile layer that is designed for export, rather than displaying live map tiles. But you need an ArcGIS Online subscription in order to use it.
It might be possible to write some Python/JSON code that accomplishes the task of exporting data this way, but perhaps not without special permissions. I think the core of your question is whether there is freely available higher-resolution data than the Earth Explorer is providing for download, and that is one which I am not prepared to answer in full with total confidence...
The availability of a resolution comparable to the tiles you are referring to (which appears to be less than 1 meter) would heavily depend on the size of the region you are interested in. It's hard to get access to recent, high-resolution satellite imagery without paying for it. You might be better off looking for local data repositories that might have obtained aerial imagery in the area you are interested in. For example, in Pennsylvania we have PASDA, which provides access to aerial imagery at differing levels of currentness for most of the state, with some resolutions as small as 0.5-foot.