A couple of days after I had posted my complaint about Aisler's lack of email response here, Felix did send a nice reply. (Not sure whether it was prompted by the post here or a coincidence.) He explained that they indeed make those changes to the drills, and stated that the Chinese manufacturers do the same without telling you.
Since I had already ordered the board in question from JLCPCB, I thought this would be a good opportunity to compare the two, and ordered the same little PCB from Aisler too. Both boards have arrived now, and I have attached closeup photos of both of them. A few comments:
Both manufacturers did a clean job on the etching; no problems whatsoever with the 5 mil traces, as far as I can see.
Aisler has indeed made the via drills larger (0.25mm to 0.35mm), without enlarging the annular copper ring. The resulting annulus is very narrow, but they seem to get away with it since the registration of drills vs. copper is excellent. JLCPCB kept the drills unchanged, and has maybe enlarged the annulus a bit (making it non-spherical to keep a safe distance from the pads). They can use the extra leeway, since the drill registration isn't quite as good. Either approach should work, I think, but it feels like Aisler is taking a high risk here, or has a lot of confidence in the precision of their drills.
Aisler has also enlarged the openings in the solder mask a bit, apparently in proportion to the pad size -- it is hardly noticeable with the BGA pads, more so with some larger discrete components on the bottom. JLCPCB has shrunk the openings for the BGA pads somewhat, distorting them to cover any tracks which connect to the pad. That may be a good idea, to avoid variations in the copper area available to the BGA's balls, but is probably not critical?
Aisler does not officially support tented vias. They did cover the copper annulus of all vias with solder mask, but some of the holes are filled/covered, while some are not. For the BGA, I feel a bit more comfortable with the JLCPCB board, where all vias are closed. (But the surface is not planar, and I realize that I can't expect that.) Not sure whether it makes a practical difference though.
Aisler's silkscreen print is much more precise that JLCPCB's. Not very visible in the photographs, but JLCPCB's ink is thicker, with less well-defined contours. But it's clearly readable at 0.8mm character height.
JLCPCB did a shabby job with the contour milling. Some corners have strange little steps or indentations; and they apparently took some sandpaper to some of the boards (badly) to cover that up. The two corners shown here are some of the more pronounced samples, not all of them are that bad. Aisler's milling is perfect, giving the slightly rounded corners I wanted. But then you get the mouse-bites on all sides which I hadn't expected...
Finally, JLCPCB does of course give you more choice. With Aisler, green soldermask and 1.6mm material are the only options you get. Worked for me in this case.
In summary: Both boards seem usable. Given the low shipping cost and the favorable price structure for small quantities, I will definitely order from Aisler again.
Edit: Forgot to mention -- I like Aisler's stencils much better. JLCPCB sent one huge sheet with sharp corners (30*40cm², for a PCB that is DIP-40 sized) with combined stencils for both sides. Aisler provided two separate postcard-sized stencils with nice rounded corners, which are handy but still have enough surrounding space to fixate them and work the rake. I will definitely use the Aisler stencils. Thickness is 100µm for Aisler and 150µm for JLCPCB, by the way.