My Rebbeim always taught me not to judge the greatest poskim, halakhic deciders, by their Chumrot, their stringencies, but rather by their kulot, their leniencies. This is because any Am Haaretz can be strict. It takes a great Talmid Chacham to find the areas where one can be lenient utilizing the halakhic process to help people in need.
One can see the source for this from today's Daf. The Gemara says: כך אמר דוד לפני הקב"ה רבונו של עולם לא חסיד אני שכל מלכי מזרח ומערב יושבים אגודות אגודות בכבודם ואני ידי מלוכלכות בדם ובשפיר ובשליא כדי לטהר אשה לבעלה. David says to Hashem, Master of the World, am I not saintly? All the kings of the East and the West sit with all their pomp among their company, whereas my hands are soiled with the blood [of Niddah], with the foetus and the placenta, in order to declare a woman clean for her husband. (Translation courtesy of Halakhah.com).
What this statement is saying is that David was willing to literally get his hand's dirty in order to permit a woman to her husband regarding the laws of niddah and zavah. The emphasis here is clearly on the importance of halakhic deciders finding leniencies whenever possible, obviously while still working within the halakic process.
Many people view Rav Moshe Feinstein Zatzal for example, the great posek of the previous generation, as a machmir when in fact the opposite is true. One notable example is concerning the status of milk in the United States. According to the Shulchan Aruch, milk is only kosher if a Yisroel watches the milking. However, Rav Moshe in a famous teshuva permitted all milk in the US based on the fact that the government inspection is tantamount to a Yisroel watching. One can read a modern day revisiting of this pask on the OU's website.
What an incredible leniency without which how many thousands of Jews would not be keeping kosher today! This is the power of great halakhic authorities. They are willing to get their "hands dirty" in order to find leniencies whenever possible.
One very important proviso is in order here. While great poskim are known for their leniencies, at the same time, they always act with a great deal of intellectual humility, operating within the halakhic framework and not distorting it to serve their purposes.
We see this from the continuation of the Gemara which says: ולא עוד אלא כל מה שאני עושה אני נמלך במפיבשת רבי ואומר לו מפיבשת רבי יפה דנתי יפה חייבתי יפה זכיתי יפה טהרתי יפה טמאתי ולא בושתי. And what is more, in all that I do I consult my teacher, Mephibosheth, and I say to him: My teacher Mephibosheth, is my decision right? Did I correctly convict, correctly acquit, correctly declare clean, correctly declare unclean? And I am not ashamed [to ask].
While King David strove to find halakhic leniencies, he did not take this as a source of intellectual arrogance Chas VeShalom. He constantly consulted with his Rebbe, Mepiboshet, whose very name indicates that he was willing to humiliate David his student in matters of Halakha. Our great sages in their striving for halakhic truth, while striving for leniency whenever possible, are not afraid to be wrong and readily admit when they are wrong even it means embarrassing themselves in front of their teacher.
This also is a hallmark of our great gedolim today. There are numerous stories of their humility and ability to admit when they are wrong. For example, there is a famous story told by Rabbi Shlomo Riskin about how the great Rabbi Yosef Soloveitchik, known affectionally as the Rav to his thousands of students, once snapped at a student who challenged him in shiur. After the shiur was over, the Rav thought for a few minutes and realized that the student had been correct and it was his analysis that had been faulty. He proceeded to walk to a restaurant across the street where the student was eating lunch in order to publicly admit that he was wrong and the student was in fact right.
This is the greatness of our gedolim from King David to the present. On the one hand, they possess the brazenness to "get their hands dirty", to seek out leniencies within the framework of halakha whenever possible. On the other hand, they balance this with the bashfulness to constantly question their own analysis with both their teachers and students. They are not afraid to humiliate themselves in the path to seeking the halakhic truth.