For an aluminum beam the steel beam calculations would give you 3x the
deflections expected from the steel design.

I would start with the deflection calculations, which, generally would
make the remaining stresses OK. They would still require checking,
though, since it is impossible, for me at least, to tell which specific
conditions apply without doing the arithmetic.

A word of caution on using aluminum: I would not use aluminum in a
hoist application without knowing the COMPLETE history of manufacture
and usage. The reason is that there is a huge variety of Al alloys,
with differing heat treatments, made for differing applications (eg.
architectural vs. structural). Without knowing all this it is
impossible to determine which yield stress to use for design.

You should also be aware that aluminum does not have an endurance
limit, ie. it WILL, eventually, fail in fatigue! That is why it would
be important to know the usage history of the Al beam. Further, was it
subjected to any heating, welding (especially attachments, mid-span)?
Because this would negatively affect the heat treated strength
SEVERELY.

I am sorry to say that I know of no safe way of using an Aluminum beam
of unknown history in a hoisting application.

Wolfgang