US immigration processes are a chancy and fast-moving target, as well as being subject to arbitrary interference from the political system. I have been peripherally involved with an "extraordinary ability" application before, and my understanding from that process is that it's awesome if you can get it, but highly unpredictable. I would thus strongly recommend against counting on EBI-1A.
On the other hand, H-1B also has its downsides: the number of available H-1B visas is capped through a highly complex formula. You also can end up "hostage" to a poor employment situation because you go immediately out of status if you quit or are fired. You also count as a non-US person, subject to export controls and potential limitations on the types of work that you can do---with some companies this is no constraint, with others it severely limits your career.
Instead, the first thing I would recommend is to get a trustworthy immigration lawyer involved to help you through the process. See if it's possible to do a "belt and suspenders" approach where you simultaneously aim for the extraordinary ability green-card (since getting a green-card gives you more personal and professional flexibility) and the H-1B route with the company for backup. Even with the company taking charge of an H-1B, it's good to have a US lawyer advocating for you also, because if a random low-level clerical mistake at the company may be "Oops, too bad!" for them but a wrenching restructuring of your life plans for you.