Many thanks, this is very interesting!
There is no native aralia in Ireland and I have never come across A spinosa. Nor have I ever seen either in the wild. here Must pay attention next time I'm in the US.
It doesn't seem to naturalise here - I suspect it may not fruit or the seed is not viable/doesn't germinate because of the relatively cool climate and short summers. Same as a lot of plants from the southern-hemisphere or warmer climates here: they survive quite well because of the relatively mild winters but don't flower, fruit, or seed.
My plant came with three stems and I wouldn't really want any branching. When it comes to pruning them back I'll try what I do with cordyline australis (grows everywhere in Ireland and is mistaken for palm) and remove all but one of the new shoots it makes below the cut.
How big are your pots?
here is a photo of mine. They are very striking architectural plants!