So are we saying that anyone can avoid UK Border checks by coming in through Eire?
UK and Ireland systems are linked for checking on immigration as you correctly point out that people can use either country to get to the other so they do know anybody turned down in one. People also use Belfast but now there are sporadic checks by Irish Immigration on trains and buses and also by PSNI at Belfast airport.
If get kicked out by UK or Ireland then pretty much you can guarantee that entry into the other one later automatically is flagged.