If you'd like to have more verification for your certificate you need a extended validation certificate (which often costs money). These certificates also include your (company) name and the issuer verifies whether it's correct or not.
Basic certificate issuers don't judge over domain names or content, they just verify domain ownership.
It is complete and total bullshit that DV SSL certs still cost money (thanks LetsEncrypt), anywhere from $9/year to $80/year.
The companies that rely on selling DV validated SSL certs for their business model are polishing the brass doorknobs on the Titanic. It's all going down. Just a question of when.
In the past, the bar used to be much higher.
Whether a lower bar is a good idea or not I will leave to other more informed folk.
For identify validation, you'd need to buy an EV or OV cert. But for most organizations, particularly if your domain IS your identity, a domain-validated cert is absolutely fine.