But it is possible to get a consistent look and feel for fonts, by using web fonts.
Web font bugs show up in the weirdest places. Unless you're constantly vigilant and manually test across all the platforms you want to support, some users will see crappy fonts. Hardly anybody goes looking for these bugs, so the bad fonts can be shipped in production for years without anybody at the company noticing. This is what has happened at Apple, Wired, and Medium.
Not that consistency is worth striving for. There's not much consistent about what you can show on both a 6" 4K OLED and a 15" 1366x768 TFT LCD. Certain fonts will look great on the former and be borderline-unreadable on the latter.
If we're talking about having the typography on user A's phone look the same as on user A's desktop, that seems a bit more reasonable, but the rest of the UI is necessarily going to be drastically different save for perhaps the color scheme.
Or is the "consistency" that matters here about making sure the website is pervasively using the same typography as the advertisements?
Well, I suppose that's a valid question, but it's moving the goalposts. We were talking about having consistent fonts, and I'm claiming it's absolutely possible by using web fonts (after all, that's the reason web fonts exist).
We still haven't figured out where the goalposts started!