Europe -> US

If the goal is to permanently move and reside in another country, however, there are alternatives that may make sense

- further education if you can afford it (employers paying for this is increasingly rare, but also possible); and then go the CPT/OPT/H1b route which may take a few years.

- date/marry someone in the destination country. This makes even more sense if you come from a community where people prefer partners from their own culture. Doesn't have to be arranged, but there is a reason why so many niche dating apps and portals exist. You may have a preference that your spouse has a certain cultural, religious or linguistic background. Or maybe you don't want to be alone any longer. It doesn't matter, marriage is a key route into a new country these days and it is one that will  work for many people for generations to come.

- family sponsorship is also possible and a straight-forward route, normally. Your first degree family members can sponsor you (i.e. parents, children, siblings)

- if the destination country is more important than the career you currently have - some people have a life-long dream of starting their own business or do something else than sit at home or commuting to the office. If you have already decided that moving to the other side of earth is the right thing, why not go one step further and also rethink your career? Now might be the best chance.

- investment based green card; possible if the amount of investment exceeds 900K (immigrant route) for immigrant intent (less for non-immigrant)

- Extraordinary abilities, O visa for professionals in entertainment, modeling, acting, singing, science, business, (..) - if a strong case can be made based on outstanding, published results (i.e. SAG/AFTRA credits, scientific awards, noble prize, published in national/international media, Olympic medals, ..). It is possible to get the O visa if you are a strong contender within your space, but not for a regular finance employee who works in an office.

Internal transfers happen all the time, I have seen them across multiple industries and companies.

As you can see, there are various avenues into a new country and many options are available. Simply applying to a job in the US and hoping to land a visa sponsorship is indeed a possibility, but I wouldn't bet my career or life on it. You would have to time the application in the right window, the employer would have to file the visa application, pay the attorney, justify the investment internally and also externally (labor market assessment), and then in all likelihood not be able to get you the visa anyway (due to the lottery of H1b). This is a very unlikely avenue, but it is not impossible.

Also, you are already participating in the the diversity lottery- keep doing that.

On the VWP you are legally allowed to enter the US at an official PoE and you are then (normally) given 90 days to exit. In this time frame you are also allowed to meet prospective employers, even interview or undergo training - just not work, or get paid. It is very unlikely that a US based employer would interview someone who doesn't hold a work authorization.
Also, the MBA (or any degree) is no guarantee for the H1b or a green card. They may make things easier to get further jobs/visas, but there are plenty intl. grads who go home in the grace period given by USCIS. The only guaranteed way for immigrant purposes into the US is to invest a large amount of money (EB5), win the DV lottery, or marry a USC/LPR.
For non-immigrant purposes there are quite a few alternatives that are easier, if all you want is to live in the US for a while and then return to Europe.