Earlier quoted context omitted.
You said it yourself: Users are viewing your site on devices with totally different screen sizes, different resolutions, on different platforms, with displays of varying brightnesses, glare, viewing angles, color spaces, and in environments of varying levels of lighting. Given these constraints, it's not possible to get a consistent look-and-feel. So don't waste resources trying. You'll create more problems than you'…
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.