I'm not sure why you think that speakers of Romance languages would use the word 'that' in English more often than native speakers. If you use the word 'that' three times in a grammatically correct sentence, it sounds fine. For example, it's quite natural to say: 'That's the girl that I met at that party'. Or you can omit the relative pronoun in this sentence if you wish, and say: 'That's the girl I met at that party.' Both sentences sound fine. What does sound wrong, though, is when non-native speakers put too much stress the relative pronoun. In the sentence 'That's the girl that I met at that party', the second 'that' is NOT pronounced in the same way as the first and third ones. The relative pronoun in the middle of the sentence is pronounced with a weak 'schwa' vowel /ə/, which sounds very different from the open 'a' sound of the 'that' at the beginning and end of the sentence.