The main benefit of the lawyer here is to break through the bureaucracy on the other end. Your lawyer talks to their lawyer, their lawyer talks to someone on their end who is actually authorized and able to make changes (not necessarily to make them change it - possibly just to figure out what is going on).

This causes a human with authority to actually look at your case and likely go "yeah, this is silly, I've taken care of it". Lawyer is very happy because something that looked like a lot of expensive work just went away.

I guess you could try to go for the tortious interference angle:

I have a contract with the customer which involves interacting with them by email. Customer uses EmailCompany. EmailCompany unfairly blocks me from emailing customer. EmailCompany is interfering with the contract. Both myself and the customer have a reasonable expectation that we should be able to carry out our contract by emailing back and forth, so EmailCompany is obstructing the contract.

Any of this sort of "advocacy" only works if the customer actually cares about receiving your emails though.

In my experience, the customer will just switch email providers when I tell them we're blocked. I had an email provider just the other day who wasn't accepting my emails. I tried to get them to unblock us, but to no avail. I told my customer and he went and registered a new email elsewhere.

Lawyers typically will only talk to other lawyers when interacting with other companies (often are ethically compelled to).

An email from legal@companyA.com to legal@companyB.com will be read.

Good point, generally. However, it depends on the list or the tactic that's used. Blacklisting an ISP's ASN or entire network range because the blacklist creator set an arbitrary threshold of acceptable number of spam activity from 1 or more IPs. I'm really not sure about the legal aspects here, but there are so many practices that blatantly approach extortion. And it's skillfully done in the name of "online etiquette" or a "safe internet".

Obviously, legitimate use-cases exist for such services, provided that they are operated by faithful people/entities with some level of credibility.

No, they aren't. This tactic isn't effective when you've truly pissed off the other party. If it could plausibly be an innocent mistake, this strategy will cheaply and quickly get things resolved.