Image Capture is coming preinstalled in your Applications folder.
From your description "I tried the obvious steps of “import all new items” and selecting all my photos and trying to import them manually." I take it, that the Photos.app did show you the connected iPhone, also could separate them into "New" photos and "Already" imported, which is good. How long did you wait, after you selected to import "New" photos? Did Photos complete the import or have you seen an error message?
You have been looking in the "Imports" album for the imported photos. Have you also checked the Recents album? It may take a while for the photos to appear in Imports, usually they will appear in recents before we are seeing them in Imports.
How much free storage do you have on your Mac? If you cannot import large amounts of photos at once, try to import just one single photo. Can you at leat import a single photo?