First it would seem you are not remembering 97:1 correctly. There it says:
Do not knead dough with milk since (1) it may come to be eaten with meat. If milk is kneaded with dough, then the bread is assur 1) to eat
even on its own. However, if only a small amount of such bread is
made, for example one meal’s worth, or if the bread had a special
shape1 that would remind you not to eat it with meat, it is mutar.
Similarly, one should not bake bread in an oven that has been greased
with fat. If bread was baked in a greased oven [it is assur] for the
same reason as bread kneaded with milk.
RAMA Therefore there is a custom to make bread with milk in it for Shavuos, also bread with shuman in it for kovod Shabbos since this is
considered a small amount 2 and its shape is different than that of
bread made during the week. Kol shecain phladan 3 or pashtaida is
mutar. 2) One should not bake bread with phladan or pashtaida in the
same oven since there is a concern that fat may leak onto the bread.
If the fat does leak under the bread it has the din of being kneaded
with fat. The custom is to put parve food in the opening of the oven
[when milk or meat are also being cooked]. (3) Even if the parve food
is in a pan the custom is to be strict l’chatchila.
You will notice from both the words of the Mechaber as well as the Rema that the worry is making large amounts of dairy bread without any sort of sign that they are dairy. Not just dairy bread. This is a technicality that every US Kashrut organization would relies upon to give heksherim to various dairy breads in the supermarket. So long as it carries and OU(D) or OK(D) or whatever(D) that in and of itself is considered enough of a siman. In Israel one can bake dairy bread so long as it is triangular(the reason for the mandated shapes of Bourekas by Rabbinut).
Now let us look at Y"D 96:1
A tznon or 1) silka5 that was cut with (1) a ben yomo 2) meat knife
or [a knife that] 3) wasn’t cleaned is assur to eat with milk unless
4) a k'dai natila, the width of the thumb, is removed from where the
it was cut (2) 5) or if it was tasted and did not taste of meat in
which case it is mutar [to eat with milk] after it is washed. (3)
V’yaish omrim 6) the same law applies even if the knife is an aino ben
yomo and clean. If the k’dai natila was not removed, nor was it even
tasted before it was cooked with milk, 7) then 60 is needed (4) 8)
against the area (5) that was cut. (6) The same din applies if the
knife of a non-Jew was used.
Here we see that for the davar charif to become "meaty" certain conditions have to be met. First the knife must be ben yomo. Second the surface must be left in tact with an actual meat taste. However, if it were to mixed with milk, even then you would only need 60x the surface area of the cut. Negating either one of those conditions would mean that the davar charif is not meaty. Unlike the bread which is impossible to make not dairy.
As for the bread the concern was that one would make a quantity of bread(more than enough for a single meal), and thus the leftover may come to be eaten with meat. For instance the minimum measure for taking challah would be enough quantity to make several large loaves, and thus in fact would be enough for several meals. Whereas with the davar charif, we are only worried about the surface of the cut, and then only if it has a discernable meaty taste after having been cut. In short it would seem that Chazal was not worried about it being prepared in quantity enough for it ever be an actual concern.
