My favorite take on this is that yes, in fact it is just the chain rule. The usual argument goes that automatic and symbolic differentiation are fundamentally different, so anything particularly old (pre-computers, for example) doesn't count as inventing back prop. But here's my favorite take on equivalences between AD and symbolic diff [0]. I wish there wasn't such importance placed on who invented it for stuff like this. Clearly, someone codifying backprop wasn't a bottleneck in making ML progress, so why's it get so much attention?

[0] https://emilien.ca/Notes/Notes/notes/1904.02990v4.pdf