"service doing business with EU customers "

If the EU customer is a company and not a private citizen and if it does involve storing personal information of the EU customer customers.

"What you're implying is that the EU can't legally have access to the majority of US-based Internet services"

No. You as an EU company can't transfer customer data to or redirect customers to US companies in a legal way.

"Otherwise, it seems like you're saying that EU data in general can't be legally exported from the EU to most of the world"

It depends on who does the "exporting" and what the "exporting" includes. But in general yes, it can't if the citizen whos data is exported can't be guaranteed to have the same rights as with the data in the EU.
That is the core of it, Facebook can't offer a website in the US that is open to the EU and take information on the website "exporting" the data by POST HTTP requests to it's servers in the US (which they don't because Ireland, but in general yes).

There is much more to it, like "Can I store customer data in Google for Business spreadsheets?"

You're probably fine with Gmail because people know that this is an US company and sending an email to an US company is something a EU citizen might want to do. It's not as clear with Fastmail. It's not clear at all if you use custom domains with both. If you use a custom domain, where the customer can't see that it's outside the EU, do you export email addresses outside of the EU e.g. to Australia? But data protection agencies in the EU will take some more time to arrive at all these finer nuances. For now they are focused on Facebook and Google, and in 2021 went one level deeper with acknowledging that it's illegal to use Mailchimp in the EU (currently, I assume Mailchimp will create an EU legal entity and host EU data in the EU in the future when pressure rises).

For large enterprise that have subsidiaries in the EU it's already illegal to transfer EU employee data to the US for processing.

"You're claiming EU residents don't legally have access to them?"

As an EU citizen you can do whatever you want with your data, so "EU residents don't legally have access to them?" is misleading, because it would not be illegal for the citizen but - if - for the company.
The company has a problem if it can't prevent three letter agencies from accessing the data. If you have no assets in the EU and do not plan on visiting the EU there is not much to fear though probably. I think it might be legal - not sure I've read something about it - to process data e.g. as a hotel for EU tourists, if you delete the data afterwards. Yes the GDPR is broad.

"But if I sell software in multiple countries, and part of my account process is collecting an email address or other PII, is that not GDPR compliant unless I set up offices in the EU?"

If you sell drugs to the US you're in trouble, even if it is legal to sell drugs in the country you live in.

You as a US company probably can sell to EU citizens but IANAL. The bigger problem for the EU if you sell to EU citizens from the US is VAT. If you send physical goods then your customer needs to pay VAT at customs - as many people in the EU found out after Brexit, if you send digital goods then the customer usually doesn't pay VAT. This is what agonizes the EU more than the GDPR and is the base for France to charge digital taxes to US companies.

"I wouldn't be hearing about it on Hackernews if that was the case."

Well I've lost 100+ karma for pointing out in the last years that it is illegal for EU companies to use Mailchimp. Today is the first discussion where people agree - still I lost 10 karma.