Homepage authority does flow to deep pages, but it attenuates with click-distance and link dilution, so a strong homepage helps and cannot alone carry pages buried many clicks deep or competing for hard terms. The honest verdict is helps-but-attenuates-with-depth. The authority is real and it does travel inward, but it weakens at every hop and every page it spreads across, so the value reaching a page five clicks down a sprawling structure is a fraction of what reaches a page linked straight from the homepage. Distance and dilution decide how much actually arrives.
Two forces do the attenuating. Click-distance is the number of links a crawler and authority must pass through to reach a page, and each hop sheds some of the strength, so a deeply nested page receives far less than a shallow one. Link dilution is the splitting of authority across all the links on each page, so a homepage that links to a hundred places gives each destination a thinner slice than one that links to a handful. A page that is both far from the homepage and reached through pages that link out widely gets very little, no matter how strong the homepage is, which is why a strong homepage is necessary support but no guarantee for deep or competitive pages.
This calibration rejects both extremes. A strong homepage does not lift everything automatically, because the authority thins out before it reaches the buried pages, and the assumption that homepage strength alone ranks deep pages on hard terms breaks against attenuation. But it is equally wrong to say homepage authority does not reach deep pages at all, because it plainly does when the path is short. The truth sits between, and the variable that decides the outcome is the structure carrying the authority inward.
Structure is the lever you control, so use it. Shorten the paths to the pages that matter, link them from the homepage or from prominent hub pages rather than leaving them several clicks deep, and build hubs that channel authority into clusters instead of scattering it. Treat depth and link dilution as the things to manage, and you make the homepage’s authority reach where you need it.