There are many cases of businesses operated on Shabbat in partnership with non-Jews. The classical case is that of a nursing home operated by a Jew, which needs to remain open and functioning on Shabbat. One is allowed to operate such a business by taking on a non-Jewish partner and stipulating from the beginning that the non-Jew is totally in charge of Shabbat work and receiving its remuneration. (Note that structuring these arrangements require a knowledgeable rabbi).
There is no need to rely on amira l'akum since one does not tell the non-Jew what to do, he does it on his own. There is also no issue with the Jew being the CEO. The key is for the non-Jew to receive the benefits made on Shabbat.
In the words of dailyhalacha
There are, however, arrangements that could be made to allow a
business to run on Shabbat if the Jew co-owns the business with a
gentile partner. Namely, as discussed by the Shulchan Aruch and Mishna
Berura (OC 245), a Jew may
arrange that his gentile partner receives all profits earned from the
business on Shabbat. For example, the partners could agree that the
gentile receives all profits earned on Shabbat, and the Jew will
receive all profits earned on Tuesday. Once they agree to this
arrangement, they may then divide monthly or annual revenues equally,
even if the Shabbat profits exceed the Tuesday profits, as the excess
revenue received by the Jew is legally considered a gift given to him
by his partner.
See also here and here for more detailed overviews.