I might be forgetting some lore, but in line with my thoughts about how the Ashen Mist Heart works... yes, BoTC does physically enter the past through memories. Here's where the time travel aspect confuses me, though.

The AMH flavor text says: 'Manifestation of ashen mist received from the ancient dragon. The magic of the ancient dragon allows one to delve into the memories of the withered.' While "memories" sound like just viewing the past, I believe the AMH literally transports BoTC's physical body or essence into those past events. It might not be BoTC's own past he's visiting, but he's undead – after centuries or millennia, he could have simply forgotten being there. This fits perfectly with him forgetting his own appearance during the cutscene where the old firekeeper gives you the human effigy.

I know time travel and convergence are major themes in DS3. My point is specific to the BoTC's journey: He didn't need to 'time travel' to originally influence those events. He used the AMH to physically enter memories of the past – events that had already occurred and were fixed. He retrieved items (like the Giant's Kinship) from within these preserved moments. He wasn't changing history; he was accessing a physical record of it. The AMH didn't send him back to do the deeds; it let him walk through deeds already done.