As I understand it, the law requires that employers be able to demonstrate that the job cannot be filled by either local or EEA nationals before they can sponsor it for a foreigner. This generally means advertising it for 4 weeks in at least two mediums before they can apply for a sponsorship, which comes with a set of regulatory and financial costs. So from the employer's perspective, it's a hassle and takes a long time to fill an opening through this route.

What that means for your boyfriend is that he's not likely to find anything through applying to jobs blindly on LinkedIn or Indeed. The better path here might be identifying companies or opportunities that have a need for his specific skillset, then making personal connections with folks in those companies who might be able to recommend him. Have you all tried networking to find specific opportunities that his skillset would match with?