They did fine even before MS was prevented from behaving anti-competitively. When Microsoft had to put its browser choice banner in the EU, Firefox was already at its peak.
Microsoft did it to themselves, IE4 was a great browser for its time. It was bundled in Windows, for free, it was anticompetitive, but its technical merits made a big part of its adoption. But as it became old, and Microsoft did nothing to improve the situation. When Firefox came out, it didn't need advertising from a tech giant. It was a godsend for developers (it followed standards!), who advertised for it on their webpages, user tried it and loved the cool new features (tabs!) and installed it on their friends computers, the whole tech world except for Microsoft rooted for Firefox because it was just so much better.
When trade commissions arrived, the job was already done, Firefox had enough critical mass that it couldn't be ignored and IE-only sites were becoming a thing of the past (except for corporate intranets, these have always been the worst). If anything, they helped Chrome more than they helped Firefox as it was just launching during that time.
Marketing alone doesn't win for browsers. It is really easy to switch browsers, all it takes is to have some semi-geek friend to come by and say "hey, try this new browser, it is just like the old one, but better", and if it really is better, that's it. That's how Firefox was winning in 2010. And look how hard Microsoft is trying with Edge, and fail, that's because Edge has nothing over Chrome, unfortunately, Firefox don't have much over Chrome either (yeah, privacy, maybe 5% care).