This game is very well designed for the casual community, however you want to define it.
Based on the OP definition, content with 'relatively high' success (i.e. at least 1/4 - 1/3 odds) using DF PUG would be CT, up to Garuda, or possibly even Ultima HM (depending on time of day server, etc). Titan HM is doable in DF, and even more so since 2.1 but the odds are more along 1/20 or even worse. Garuda EM is also doable in DF with roughly the same likelihood as Titan HM; Titan and Ifrit EM - I have no experience in any format, but this just looks horrible using DF alone.
Party finder PUG dramatically increases the likelihood of Titan HM success. Also with PF, you can do T1-T2 easily, and T4 is feasible too.
I would say this is applicable even if you include other elements commonly associated with 'casual' - i.e. lack of a 'high end' gaming system, limited time commitment, absence of preternatural finger dexterity, less aggressive mindset, etc.etc.