DNA polymerase III adds nucleotides in the $5^\prime \rightarrow 3^\prime$ direction because it can only add nucleotides to the $3^\prime$ end of the previous nucleotide. This is why it requires a primer. However, does DNA polymerase I operate by the same criterion? Does it require a $3^\prime$ end of a previous nucleotide in order to bind successive DNA nucleotides?
If it does, then how can it do so for Okazaki fragments when each Okazaki fragment is unbonded to each other? It is the DNA ligase that finally catalyzes the phosphodiester linkage between the $3^\prime$ end $5^\prime$ beginning of two Okazaki fragments, isn’t it?
If it does not, then what’s the deal with telomeres? After each DNA replication event, the DNA gets shorter and shorter at the very ends because that final primer can be removed but not replaced by DNA via DNA polymerase I, correct? This suggests to me that DNA polymerase I requires a previous nucleotide's $3^\prime$ end to work with, and it has confused me regarding its action on Okazaki fragments in conjunction with DNA ligase.