I don't think it truly is a "catch 22" because many Haitians that do have a visa don't have the correct one. A few months ago there was a scandal of thousands of Haitians with a student visa, but the amount of Haitian students registered in Dominican university is much less than they amount of student visas issued to Haitians in the various Dominican consulates in Haiti. A similar situation exist with tourist visas, where most of the Haitians with one aren't true tourists. In fact, many are permanently living and working in the DR, when the tourist visa doesn't allow any of the two. If a foreigner wants to legally permanently live and work in the DR, they must have a resident visa and not a tourist one.
Many of the Haitians that have one type of visa but are doing things that the visa isn't for would claim that they are legal in the DR and while holding those visa means they are technically legal, having a student visa doesn't mean the Haitian is an actual student and having a tourist visa doesn't mean the Haitian is an actual tourists. Things like resident visas are renewed in the corresponding office in Santo Domingo, no need to leave the country for that.
There is another issue affecting Haitians and I think this is simply ignorance affecting them more than an actual issue from the system. This one is that many Haitians think that by simply having a document "proving" their legal presence in the DR, thst it automatically meansthey are legally in the country. They tend to ifnore things such as whether the document expired because it wasn't renewed or even if the document is a real one and not a copy of a real one that they bought in the streets. While for them possessing these things means they are legal, in reality it doesn't. That causes an issue when they are caught by authorities, because they claim they are legal and still deported, but the document id either expired or false. They seem not to accept that expired or false documents aren't valid.
Many of the Haitians with the incorrect visa or with an expired or false document usually benefitted from corruption as in all those instances they bought them (includes paying a bribe), but didn't truly qualify with the requirements to get them.