Employment based visa

Yes you would be trying to cheat the system. Sponsorship means you would have to stay with that employer for anything from 5 years or more. You cannot just up and leave if it does not suit you. You would be more than likely sent back home if you did not stick to the terms of the sponsorship as it is in effect, a contract.  The sponsor is legally and financially responsible for you so if you get into trouble of any sort he has to take care of it. So you can see why you cannot just use sponsorship to get into USA and then do what you like. 

There is a paperwork trail that has to be completed too.

Getting a green card allows you to work in USA but it does not give you protection from deportation, the right to vote, or access to government benefits.

Once you have the green card it takes 5 years before you can apply for citizenship.

If you are in the United States on an H1B visa (work visa) and leave a job you have 90 days, depending on circumstances to find another employer willing to provide sponsorship. Sponsorship is hard to find right now, actually next to impossible unless it is a Fortune 500 company that is will to go through what it takes to do it.

That said, since you are a UK citizen and do not have to apply for a B2 visa and qualify under the VWP, you are legally able to enter the United States and look for work - just so long as you do not actually WORK. If you were on a B2 visa, you would be obliged to tell the VO (visa officer) at the embassy or consulate abroad that you are going to visit the United States with the possible intention of looking for work in the future.

If you enter the United States on an H1B visa, quit your job, do not find another employer that wants to sponsor you, you are required to leave as determined on your I94 and H1B document as stamped by CBP and as directed by USCIS. That said, if you enter the United States, work for said employer for 3 years most likely up to 6, THE EMPLOYER may file for PERM Labor Certification, I-140 Immigration Application and finally the AOS (Adjustment of Status). None of these steps are self-sponsored. Therefore just because you are employed does not mean you automatically qualify for permanency - your employer drives it, not you.

In your case, you would be under the EB3 employment based immigrant visa (see - Employment-Based Immigration: Third Preference EB-3) which is a lower priority Immigrant visa category.

I responded to your private message earlier.

There are several agencies that regulate these processes that I mentioned and you talked about - CBP, ICE  and USCIS. All of which are not agencies to mess around with - they report cross-border to the UK, Canada, NZ, Australia the European Union, just to name a few. Therefore if you violate immigration law in the United States the violation is reported elsewhere as well. Other countries such as Brazil for example find immigration law to be a civil matter. Here is a felony criminal matter.

Thank you very much for your interesting narration & parting words. It is clear to understand for me.

"adjusting to permanent resident"- by this I assume you mean a green card? Getting a green card through your employer is a lengthy and expensive process. No employer is going to do it for a lukewarm, lackluster employee, and it's a long enough process that I expect you wouldn't want to be in a job you don't really like for the duration either.

I'm also wondering how you think you'll be hired for jobs outside of your expertise or unsuitable in the first place. Employers are only going to hire someone requiring a work visa if they can't find someone domestically that have the relevant skills.

One question though, what is this website you found that lists jobs that are sponsoring work visas? I know it's come up as a question thread before so it would be nice to share this information for the benefit of other job-seekers.