That link also mentions the requirements around an ex-offender policy that the company needs to have. I'd request that if I was declined, to ensure the employer is actually following a process. If not, there could be some room for negotiation.
Smaller organizations that are not in regulated industries will be better bets for the original poster. Finance, pharma, healthcare, education etc. are all regulated industries and will have to maintain some level of compliance. How well they do it will vary, but if you aren't flagged initially, you may well be flagged later on. This happens all the time. Small FinTech specialises in trying to skirt the boundary of compliance at times, so you may find that some are laxer than others.
A sweet spot might be software efforts supporting manufacturing or marketing. There are a lot of agencies that build smaller software for large organisations that don't do these in-depth checks and are often building/demoing in their own environment before handing everything over to the company or providing the hosting so that a department can run shadow IT.