Why was not RNA completely replaced by DNA when the RNA world evolved into a DNA and protein world?
A related question is Why do some bad traits evolve, and good ones don&#39;t? Basically a strong selection pressure is necessary to filter out a trait.

Having said that, there is no strong argument in support of transcription being disadvantageous.

RNA has unique properties that makes it suitable for certain functions. RNA can adopt different secondary/tertiary structures that an equivalent ssDNA cannot. One of the reasons is that GU wobble pairs are stable in RNAs but not in DNA thereby allowing structural flexibility to the former. See https://chemistry.stackexchange.com/a/35224/5295

RNA also has a high turnover because of different RNAses. If there was DNA instead of RNA for carrying temporary messages and there were equivalent DNAses to increase the turnover of the "mDNAs", then the genomic DNA would also become susceptible to degradation by these DNAses, which would cause genomic instability. You can argue that protein synthesis could directly begin on DNA as a template instead of mRNA. I guess that it would crowd the DNA and clash with replication. Even transcription machinery clashes with the replication machinery; this is called Transcription Replication Conflict which leads to genomic instability. A hypothetical situation in which there are multiple ribosomes on a DNA, is likely to have higher chances of the conflicts with replication than transcription-replication conflicts, thereby causing higher degree of genomic instability. I am assuming so because a large number of ribosomes need to simultaneously be progressing through the gene in order to produce sufficient amount of corresponding protein (like in case of polysomes).

Another possibility is based on the RNA world hypothesis. If it is really true that RNA preceded DNA as a functional as well as a hereditary molecule then it is difficult to get rid of all RNAs that easily. Even if most of the functions of the archaic RNAs were taken over by DNA and proteins, they are still performing some vital functions. To replace RNA from these roles, an alternative should exist which performs much better than the RNAs.

These are just guesses. An absolute reason can not easily be attributed to many biological observations. This is because we do not know the actual evolutionary trajectory of how things happened. However, what is important to keep in mind is that evolution does to proceed towards some kind of global optimum. Evolution does not "aim" for perfection.