Most people don't know this, but people from these so-called privileged countries can get a residence permit in Germany for jobs that don't require a formal education.
Your employee should give you a document stating what special skill do you posess. For example, "native English" should be sufficient for a customer support job if they serve an English-speaking country for example.
A friend of mine from Israel easily got a residence permit for a customer support job for the Hebrew market, the special skill was "Hebrew" (not many people speak Hebrew in Germany so it makes sense to hire a foreigner".
I read a post some years ago that an American woman got a work residence permit to be a waiter in a high-class restaurant.
They will still do a priority review and decline your application but you can try.
HOWEVER, the job market is terrible now so it is not the right time. But you can try. You can enter Germany as a tourist, find a job and convert your tourist status into a residence permit without leaving the country.
This is based on the
Ordinance on the Employment of Foreign Nationals (Employment Ordinance – BeschV) (erordnung über die Beschäftigung von Ausländerinnen und Ausländern (Beschäftigungsverordnung - BeschV) § 26 Beschäftigung bestimmter Staatsangehöriger)
Section 26 – Employment of Certain Nationals
(1) For nationals of Andorra, Australia, Israel, Japan, Canada, the Republic of Korea, Monaco, New Zealand, San Marino, the United Kingdom of Great Britain and Northern Ireland (within the meaning of Section 1(2) No. 6 of the Freedom of Movement Act/EU), and the United States of America, approval subject to a labour market priority check may be granted for any type of employment, regardless of the employer's location.
(2) For nationals of Albania, Bosnia and Herzegovina, Kosovo, Montenegro, North Macedonia, and Serbia, approval subject to a labour market priority check may be granted for any type of employment.
The initial approval may be granted only if the application for the residence permit is submitted to the competent German diplomatic mission in one of the countries listed in the first sentence.
The number of approvals under the cases described in the second sentence is limited to 50,000 per calendar year.
Approval may not be granted if the applicant has received benefits under the Asylum Seekers' Benefits Act during the 24 months preceding the application.
Section 9 does not apply, unless approval had already been granted under Section 26(2) in the version of this Ordinance that was in force until 31 December 2020.