Mozilla doesn't build Firefox (which is what I assume you're meaning here by "building a web browser.") Firefox — and all those system-integration features you're mentioning — are developed by Firefox Corporation.
Mozilla is a separate company, with separate employees, that does everything except building Firefox.
(This division was originally created to firewall off the corporate moneys donated to, and influence of corporate developers on, Firefox, from the rest of Mozilla. Back when Google was sponsoring Firefox to use Google as its search engine, their money went exclusively to Firefox Corporation, with none of that money ever going to Mozilla.)
Note that this doesn't mean that Mozilla doesn't build a web browser. Mozilla develop the engine (Gecko/Servo) that goes into Firefox; and they also co-develop some of the Gecko/Servo-based browsers used by various FOSS projects — Tor Browser, for example. This is roughly the same structure as how the open-source Chromium project (which produces its own "Chromium browser") is the basis for the closed-source Google Chrome.
Interestingly, this means that for Firefox Corporation to get features into Gecko, they have to submit PRs "upstream" to Mozilla, who might very well reject them as "only serving corporate interests at the expense of the user." This is quite unlike Chromium, where both Chromium and Chrome are ultimately steered by Google.
Firefox developers work for Mozilla Corporation. Mozilla Corporation is a subsidiary of Mozilla Foundation. Mitchell Baker is CEO of both.
In 2005 they said a for profit subsidiary was for revenue flexibility. Where did you learn it was to limit corporate developers influencing Mozilla Foundation?
Google's sponsorship of Firefox continues. Most of Mozilla Foundation's revenue comes from Google through Mozilla Corporation.
Mozilla Corporation employees develop Gecko. They developed Servo until it was transferred to the Linux Foundation. They land code with no sign of Mozilla Foundation review routinely. Can you show a PR Mozilla Foundation rejected as "only serving corporate interests at the expense of the user."?
Mozilla is a separate company, with separate employees, that does everything except building Firefox.
(This division was originally created to firewall off the corporate moneys donated to, and influence of corporate developers on, Firefox, from the rest of Mozilla. Back when Google was sponsoring Firefox to use Google as its search engine, their money went exclusively to Firefox Corporation, with none of that money ever going to Mozilla.)
Note that this doesn't mean that Mozilla doesn't build a web browser. Mozilla develop the engine (Gecko/Servo) that goes into Firefox; and they also co-develop some of the Gecko/Servo-based browsers used by various FOSS projects — Tor Browser, for example. This is roughly the same structure as how the open-source Chromium project (which produces its own "Chromium browser") is the basis for the closed-source Google Chrome.
Interestingly, this means that for Firefox Corporation to get features into Gecko, they have to submit PRs "upstream" to Mozilla, who might very well reject them as "only serving corporate interests at the expense of the user." This is quite unlike Chromium, where both Chromium and Chrome are ultimately steered by Google.