It has very short half life, there’s not a ton of it and it’s likely to be wrapped around a nucleosome. Uniformly, sequences in blood are quite short (<200bp) which is far too small to code for anything meaningful (for reference COVID genome is 29kbp, and the spike protein version in the vaccine is about 4kbp). The bloodstream shreds free DNA pretty effectively, but yes, some short DNA can be detected. That’s a good bit different than a large amount of mRNA/DNA floating around though, especially if it’s a long sequence.

Useful reference:

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4715266/