I'll address the three sub-questions individually, as a way of fully answering the title question.
While there are indeed three planets orbiting PSR B1257+12, note that it's a pulsar, the compact remnant of an energetic event involving the system's progenitor. However, that event would likely have destroyed any planets that originally orbited the star, meaning that the three planets we see today likely formed afterwards (see e.g. Podsiadlowski 1993). By contrast, it seems that WD 1856+534 b may have formed prior to its parent star becoming a white dwarf, after which its orbit was disrupted and it moved into a tighter orbit.
While there are other candidate exoplanets orbiting white dwarfs, some consist of planetesimals, asteroids or other debris (e.g. WD 1145+017), some have only been detected indirectly (e.g. WD J0914+1914), and some are being actively disrupted by the star (e.g. WD 1145+017). On the other hand, WD 1856+534 b seems to be somewhat stable and is quite substantial $(M\sim10\text{-}15M_J$), likely falling into the giant planet regime.
It's certainly true that we don't know that the planet formed before the progenitor became a white dwarf, and it definitely can't be ruled out. The authors (Vanderburg et al. 2020) don't explicitly discuss a capture model, but they do attempt to make a statistical estimate of how close a nearby star is likely to have come within the past ~6 billion years, i.e. since the star became a white dwarf:
$$D\sim\left(\pi vt_{\text{cool}}n\right)^{-1/2}$$
with $v$ the mean local stellar velocity, $t_{\text{cool}}\approx5.85\;\text{Gyr}$ the cooling age of the white dwarf, and $n$ the local stellar number density$^{\dagger}$. Assuming $v\approx60\;\text{km s}^{-1}$, they find that in the last ~5.85 billion years, the closest a nearby star would be likely to come to the system is roughly 600 AU. If we consider the capture model, an encounter at that distance seems unlikely to transfer a planet to a comparatively tight orbit.
That said . . . we obviously can't rule out that sort of interaction. The same holds for a bunch of other proposed scenarios which call for "either
finely-tuned or a priori unlikely initial conditions", as Vandenburg et al. put it. So, sure, capture is possible. It just would presumably require a fairly close approach. The authors also don't (I think) explicitly address the idea of formation after the white dwarf evolved to its present state.
All of that said, yes, this isn't an enormous leap into unexplored territory, but it does seem that we have solid evidence of a massive exoplanet in a stable, close orbit around a white dwarf, and models involving it forming prior to the progenitor's evolution to a white dwarf phase appear to hold water.
$^{\dagger}$ As the system is only $\sim25\;\text{pc}$ away, they make safe assumptions about the values using data from Gaia in the solar neighborhood, i.e. $n\approx0.1\;\text{pc}^{-3}$.
