I think we have to imagine a colour space ( sRGB or aRGB ) as a cupboard, the larger the cupboard the easier it is to fit large images into it without the sleeves or tails catching on the sides or bottom and creasing. Rendering intent is rather like deciding how the bits that are too big to fit into the cupboard are folded to avoid creasing. Picture the three dimensional space withing a huge free standing cupboard, and consider it as allocated space by a matrix of x, y and z axis.
Now we have an object to hang in that space in a uniform and predictable way. If we give the object its own location identity and match that to the matrix given to the cupboard, its location within that space is both accurate and predictable, how much space surrounds it depends on the size of the object and if it reaches the edges of the space. A small object ( sRGB image) hanging in a large (Rec 2020 ) cupboard will swing untouched by its surroundings . The other way round , an aRGB object hanging in a sRGB space will require the saturated areas that are out of the spaces holding to be folded in such a way as not to exceed its space ( out of gamut handling based on rendering intent)
So now we have to take this beautifully placed object from our carefully planned cupboard , and go with it on holiday to another place. How can we ensure that it will hang so correctly in the next space, not creasing on the sides in a squash. Obviously if the owner of the place we are visiting has the same plan for their cupboard our object will hang perfectly as it did at home..... however life is never easy, their floor is uneven and leans to the left, meaning with gravity our object will touch the sides even though both cupboards have the same space inside. So now we measure the difference between the angle of the space in the holiday cupboard and build a change log of coordinates. Now the object hangs in the holiday space , as it hung at home.... the coordinates have changed or been translated to the new space ( icc profile ) but the object is unchanged in it relationship to its planned environment and has the same size as it always had ( gamut).
Now if every cupboard in Chrisindon had a similar plan given to them , our problems would be just producing a translation ( icc profile) for each and every cupboard used and our hanging nightmares are over for ever.
If an object is planned to fit into an aRGB space , this does not mean its colours have all of the colours of that space, it just means it is planned so that it will hang perfectly in that space. Planning the same object to hang in a Profoto space does not give it more colours, it just means it hangs with more spare room around it .