Know of a visa advisor in Chicago?

That's really bad news then.   

I cannot find where it states that the income must be earned by the British Citizen.  Not saying that isn't the case, but can any of you guide me to where that is?

No problem.

Appendix FM Section 1.7,  5.2 page 21

"First, the applicant’s partner must be in employment at the date of application and have been with the same employer for at least 6 months prior to the date of application. They must have been paid throughout that period of 6 months at a level of gross annual salary or income"

and here's the link:
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/420154/Appendix_FM_Annex_1_7_Financial_Requirement.pdf

The whole thing where the applicant's income means nothing is basically a theme in all the routes unless the applicant (you) are currently in the UK at the time of application with permission to work (a Tier 2 or Tier 4 visa, basically), in which case, the applicant's income can also count.

I'm sorry.   

Will you be selling a house and be able to have the £62,500?  If you have that from the sale of a house then it does not need to have been held for 6 months.

They essentially told me to get the visa and then they can hire me on as getting a work-sponsored visa is not an option for my position.  

Can you find a job on a higher level so that you do qualify for company sponsorship? Although they will need to prove that they can't find anyone else in the UK who can do that job.

Basically the UKBA will not count any savings below 16k. Then since your visa lasts 2.5yrs they want to see the amount you are short per year multiplied by 2.5 on top of the 16k.

That's the clearest way I have seen that explained.

The 16k prevents people claiming income based benefits as most income based benefits; soon to be all with the ending of the the Tax Credits benefits; are not paid if they have assets of 16k and over.

I'm sorry.   

Will you be selling a house and be able to have the £62,500?  If you have that from the sale of a house then it does not need to have been held for 6 months.

We do have a home here in Chicago that we could sell, but it was supposed to be our nest egg.  It may be one of our only routes.  I'm going to try to talk to the (potential) employer again to see if we can't me onto a work sponsored visa--but I'm afraid they are fairly certain that is not possible.  The only other thing I can think of is to help my wife find a job and get an offer letter using some of the contacts in her field.

We do have a home here in Chicago that we could sell, but it was supposed to be our nest egg.

That equity needs to be in your account before you apply for a visa. Will they hold that job for you?

I'm going to try to talk to the (potential) employer again to see if we can't me onto a work sponsored visa--but I'm afraid they are fairly certain that is not possible.  

A sponsorship licence holder knows what jobs they can sponsor. The UK only offer about 20,700 of those visas per year and they are quite strict now on the level of the job, pay, proving there is nobody else who can do that job ect.
https://www.gov.uk/tier-2-general/overview

The only other thing I can think of is to help my wife find a job and get an offer letter using some of the contacts in her field.

She would need to have a job in the US earning at least £18,600 AND a contract for a job in the UK within 3 months of her return earning at least £18,600, for you to move to the UK with her.

OR

she would need that job in the UK earning at least £18,600 for at least 6 months, before you could apply to join her in the UK.

An update:

I spoke with the relocation team of the employer, relayed my (potential) difficulty with meeting the requirements, and they told me they would take some info (which I provided) and are doing an assessment for applying for a work visa for me.  wow. Did not expect that, but I'm relieved--I just hope that they complete the entire process, not just the assessment.

Thank you all for your input.  I'll keep you posted if you are at all interested.

Does the company have any branches in Europe, say Ireland. You could try the "Surinder Singh" route. You have to satisfy the immigration requirements for say Ireland and live there for probably up to a year then transfer to the UK (you have to show you didn't move as a stepping stone). It's a lot of messing about though with two kids but I'm just bringing it up as a move of desperation.

Something to look into.

Do you have a 401K or IRA with the equivalent of 62.5 K.

I used these as proof of savings.

There was no issue.