1. I haven’t done a study, but presumably it’s not close to half-and-half; rather, say Beit Shammai are stringent in 8 out of 10 disputes. If Beit Shammai were drawing stringency with a 50% chance, then the chance they would draw stringency in 8 disputes or more is about 5% (and out of 100 disputes, the chance of drawing stringency in 80 or more is already truly negligible).

2. A tendency toward stringency would seemingly apply only in disputes over rabbinic decrees, about how far to distance things. But in a Torah-level dispute, in each case they explain that they disagreed over a certain reasoning, so what place is there for a tendency toward stringency? For example, mistaken consecration is valid for Beit Shammai, or the resting of one’s utensils on the Sabbath is, according to Beit Shammai, Torah-level.

3. Maybe in a few cases it’s hard to define what is a stringency and what is a leniency, but usually (at least as a conjecture, it seems to me) it isn’t open to two interpretations. Even in monetary law, where stringency for one side is leniency for the other, it still seems that if Beit Shammai make a guardian liable if he intended to misappropriate a deposit, that is sheer stringency (or in your view is that not a stringency?)

1. Those are the possibilities: either a tendency toward stringency, or chance. Or else the definition of stringency and leniency is not really unambiguous.
2. If a tendency toward stringency is irrelevant, then it also isn’t a stringency. If it is a stringency, I don’t see why there couldn’t be a tendency toward stringency even in Torah-level matters. For example, when there are grounds to interpret the verse this way, even if there are other grounds as well, they tend to do so, and therefore in their view it is Torah-level.
3. In monetary law I don’t see why this is a stringency. The question is whether one is looking out for the guardian or for the depositor.

1. Maybe one can also raise the possibility that the direction is the opposite. There were two groups among the people: one very meticulous and deeply committed to the commandments, and one more like the ordinary folk out in the fields. In that kind of situation, a rabbi who held a significant stringency would get pushed to live among and teach specifically the more committed group. And likewise, a rabbi who held a significant leniency would get pushed to live among and teach specifically the less committed group ("a leniency that the committed public cannot bear").
Even today this happens to a large extent: the public takes on rabbis in accordance with the general line of their rulings. So it is obvious that the rabbi of the Edah HaHaredit will be more stringent than the rabbi of Tekoa. And if those rabbis significantly changed their minds to something else, it is quite possible they would be removed by the community, rather than the community following them.
Of course, sometimes the more committed group also adopts leniencies (for example, nowadays many Ashkenazi Haredim are "lenient" about wigs), and sometimes the less committed group also adopts stringencies (for example, nowadays many Sephardim are careful to eat glatt Beit Yosef meat following Rabbi Ovadia, even if they are not among those who inspect a lulav at great length before buying it). But still, in general, the committed group places at its head rabbis whom it knew to be more stringent in practice. And vice versa. A hint to this is in Bava Metzia: "Rabbi Tarfon ruled in Lod: overcharging is eight silver ma’ot to a sela, a third of the transaction. The merchants of Lod rejoiced. He said to them: one may retract the entire day. They said to him: let Rabbi Tarfon leave us as we were, and they returned to the words of the Sages."
If the explanation is technical in this direction, then it’s not that the rabbis have a tendency toward stringency; rather, the public has a willingness or unwillingness to absorb stringencies, and a rabbi whose views happened to fit one of the publics ended up there. And the definition of stringency and leniency is unambiguous, and the skew is not chance. A balanced rabbi who is stringent in about half the cases and lenient in the other half ends up caught between two stools, with no public and therefore no students and no funding, and so his teaching sinks from view. In the end what we receive are two groups strongly distinguished along the axis of stringency/leniency.
What do you think?

That is only an evolutionary explanation, but after the evolution there is still a person there who has a tendency toward stringency.

A tendency toward stringency in the sense of willingness (and enthusiasm) to bear a heavy yoke of commandments, not a tendency toward stringency in the sense of thinking that the Torah truth is especially stringent.
And that is already a familiar and well-known human tendency that characterizes communities—the degree of inner willingness to bear a heavy burden, and even the enthusiasm to be extra careful in some area with one more clause in the Shulchan Arukh—and from there on one can give an evolutionary description of the camp-like picture in the Oral Torah.

Everything depends on how you define “stringency.” I didn’t get into that here; I only said that there is a tendency toward stringency here.
But in my opinion, a tendency toward stringency in the ordinary sense is also possible. There can be a theological outlook that the Holy One, blessed be He, wants to make things harder for us or demands a heavy yoke from us for some reason. What people call (= what I call): It is too good to be Kosher

A thought occurs to me (maybe trivial and maybe nonsense).
The longer the exile goes on, believers—sons of believers—stand there distressed that the Messiah has not come. And their intuitive conclusion is that apparently one has to be even more meticulous in order to advance and merit redemption. A feeling of guilt that gets resolved by increasing the burden, so that maybe this time we will reach the goal. Therefore the trend toward stringency keeps deepening as the exile lengthens.
There are other believers who see the current state as a kind of redemption—that is, that the repair has generally been completed and now we need to push forward. Such an approach can also be stringent regarding commandments (for example, refraining from a prohibition), but it may also yearn to add commandments such as ascending the Temple Mount and wearing tekhelet, and in that way push the wagon further forward.
And there are believers who are comfortable with the existing situation, and therefore are not eager either for extreme stringencies or for extreme leniencies.

3. You wrote that Beit Shammai’s view, which makes a guardian liable if he intended to misappropriate, is not a stringency because it assigns the money in favor of the depositor rather than in favor of the guardian.
So according to your approach, you define stringency and leniency by the number of behavioral options open to us. And in monetary law, where this is a zero-sum game, the concepts of stringency and leniency are not defined.

It seems to me that one can understand the concept of stringency more broadly, less formally, and more flexibly, so that Beit Shammai’s opinion would count as a stringency. Liability for merely intending to misappropriate creates a meticulous and exacting society in which people are very careful even about problematic thoughts. In practice it places a behavioral burden on people. Exemption from liability for merely intending to misappropriate suits a more relaxed society. The question is how much burden Jewish law imposes on the public and how bent under it they are. It seems to me that this is a more common understanding of the concept of stringency and leniency.

One can define it that way too, of course. Although it is accepted among later authorities that there is no leniency or stringency in monetary law, and that indicates that they defined it as I do.