Way back over 10 years ago we discussed the difference between the two words the Torah has for "maybe" -- פּן vs אולי. The word פּן is used when the outcome is not desired. When you see פּן expect the English translation to be something like "lest..." The word אולי is used when the outcome is desired. HaKsav v'haKabbalah writes that we know Eliezer was secretly hoping that his mission to find a wife for Yitzchak would end in failure and Yitzchak would marry his daughter because Eliezer says to Avraham אוּלַי לֹא תֹאבֶה הָאִשָּׁה לָלֶכֶת אַחֲרַי (24:5). He doesn't use the word פּן , i.e. "What will happen if/lest the girl not come with me," but rather the word אוּלַי because that is what he wanted to happen. Similarly, Yitzchak argues with his mother than he cannot impersonate Eisav because אוּלַי יְמֻשֵּׁנִי אָבִי (27:12). Yitzchak does not say פּן, "lest he be discovered," because in his heart of hearts he did not want to fool his father and he wantde to be discovered.
Given that background, let's turn to our parshsa. According to one opinion in Chazal (Makkos 11a) there is a mitzvah for the goel ha'dam to pursue a murderer and avenge the blood that was spilled. Why then, asks R' Yosef Shaul Nathanson , does our parsha use the word פּן in the pasuk פֶּן יִרְדֹּף גֹּאֵל הַדָּם אַחֲרֵי הָרֹצֵחַ כִּי יֵחַם לְבָבוֹ וְהִשִּׂיגוֹ (19:6) and not the word אולי? Isn't having a goel ha'dam something we should want and encourage if he is fulfilling a mitzvah?
Malbi"M in his peirush reads the pasuk in a way that gets out of the problem:
פן ירדוף גואל הדם והשיגו – ר״ל פן ישיגו והכהו נפש. ומלת פן אינו בא על הרדיפה, שהרדיפה מצוה לשטת הספרי
I'm not sure I understand what he is saying. Is the mitzvah only the chase? When the goel ha'dam catches the guilty party, isn't the mitzvah for him to carry out the penalty of misa?
R' Yosef Shaul Nathanson offers his own answer based on a yesod we've discussed before:
Ramban in P' Lech Lecha asks why it is that the Mitzrim were punished for enslaving Bn"Y when Hashem had told Avraham that his children would be punished by being forced into servitude. The Egyptians were fulfilling the nevuah, the ratzon Hashem! Ramban explains that ain hachi nami, had they done so lishma, that excuse would work. However, the reality is that they enslaved Bn"Y for their own ends alone. Without the l'shem shamayim, it's not a kiyum of ratzon Hashem -- it's abuse and torture. The Chofetz Chaim similarly explains that when Shaul haMelech failed to kill Agag, the navi Shmuel accuses him, "VaTa'as ha'ra b'einei Hashem," of doing wrong. You would think that Shaul's sin was in NOT doing what he was supposed to -- killing every member of Amalek -- not what he actually did do. Yet, that's not what the navi tells us. The sin is "VaTa'as," what Shaul did, because once he allowed his own cheshbonos and thoughts of right and wrong to enter into the equation, he was following his own agenda, and not purely the ratzon Hashem. To wipe out a nation based on your own agenda is genocide, not a mitzvah. We also find by the cheit ha'eigel, before the Leviim took up arms Moshe told them, "Mi l'Hashem ei'lei," (32:26, see Netziv), as only those motivated l'shem shamayim had a right to participate. You see from these sources and more that even when there is a heter, even when there is a mitzvah to cause harm, it has to be done carefully for the right reasons.
He bases the same idea on a SM"A (CM 425) that writes that the permissibility of killing a rodef to save a nirdaf applies only when the person intervening acts l'shem shamayim. If he has an axe to grind against the rodef and steps in for his own ulterior motive of settling some other score, that is not allowed.
Coming back to our pasuk, the Torah not only to tell us that the goel ha'dam was in pursuit, but also WHY he was in pursuit: כִּי יֵחַם לְבָבוֹ. Netziv comments that the Torah is just telling us the metziyus. The reason why the goel ha'dam is able to catch up to the person he is chasing is because he is in hot pursuit, כִּי יֵחַם לְבָבוֹ. When you are motivated, you have an advantage. שאע״ג שהרוצח נס ובורח במהירות, מ״מ קרוב הדבר שהגואל הדם לא ייעף ולא ייגע בחמימותו עד שישיגו Since according to Netziv it is not a din but just a metziyus, it still begs the question of why mention it.
According to the Divrei Shaul, the pasuk is in fact telling us a din. The pasuk is talking about פן ירדוף, something we don't want to happen, because we are dealing with a situation of כִּי יֵחַם לְבָבוֹ, a goel motivated by his own lust for revenge and not l'shem shamayim. There is no kiyum mitzvah in revenge for revenge's sake.