Yes, if an employer sponsors a working visa - that simply is not going to happen as you do not have a skill that is required that the US does not already have in surplus. You do not need a green card to be employed in the US. You could for example work for an employer in the US on secondment from the UK to their office based in the US.
You are not eligible for Green Card Lottery as a UK citizen, whatever your racial background - it is a programme to increase ethnic diversity and targets primarily as mentioned above countries with low immigration to the US.
Background checks do not generally require living and working in the US however - I assume however that some like Fedex may require it as other juristictions do not have to hand over such records, as the disclosure checking here in the UK has proven hence allowing foreign criminals to potentially go undetected.
In reality you would only work there in this economic climate on secondment from a UK airline or on temporary wet lease.