I am wholly new to phage display and biopanning. I am looking for a very straightforward explanation as to why suppression of termination of translation is important for phage display.
It entirely depends on the system you are using. Sometimes the phage being used carries an amber stop codon and requires suppression for propagation. This could be to restrict possible spread outside of the lab since those phages would be be able to replicate in nature. Other phage display systems rely on incomplete suppression of a stop codon so that some of the tail protein is made with the display moiety and others not.
However if your phage does not have an amber stop codon in an essential gene and your system does not rely on partial suppression, then you don't need to use a strain with an amber suppressor. But there is no downside, lots of lab molecular strains carry the amber suppressor and there is no problem having it in the strain background.
In most phage display systems there is an amber stop codon between the coat protein and the displayed protein. In an amber non-suppressing background, the stop codon is active and the displayed protein is not expressed. In an amber suppressing bacterium the coat protein is expressed as a fusion protein with the displayed protein, because of the inactivity of the stop codon. You can therefore choose whether the phage has a displayed protein or not using different bacterial strains.
During mutagenesis, you can statistically expect one of the three stop codons to be introduced for every 4^3 / 3 = 21.3 random mutations you introduce. So if you mutate enough, then your ORF is going to be under pressure to converge on that length. If you want longer ORFs, then your functional selection has to overcome that pressure.
Take out one of the stop codons, and your default random ORF length grows to 4^3 / 2 = 32 bp. That's better. It leaves you with more evolutionary power to focus on selecting for functional longer ORFs.