Its illegal to do so, according to the EULA, yes. And its not possible to login an alpha, with your omega or another alpha running. Eve will not allow it.
But people have proven there are multible ways of getting around this, if you really want. It will get you banned if you get caught, but that doesnt stop some people, you know.
I think the biggest concern with giving alphas acces to mining vessels, is the effect on the economy.
With the PLEX and moon goo chances, its pretty safe to assume CCP would like to raise prices on items, t2 items in particular. Otherwise they would have seeded more moon goo in the new minerals, some t2 stuff have almost increased twofold in price.
So it might be the direct opposite of what CCP is trying to achieve.
Its all about supply and demand. Right now thats in a pretty good place, because of alphas helping to blow crap up and the new injection of players and new accounts for old players. People can earn money on industry and manufacturing now. That might change if all alphas could fly barges and exhumers, because there will be an increase in minerals mined and a decrease in minerals used. How much, i cant say tbh.
But it doesnt take much to manipulate the economy.