In my understanding, I will split the definition of flexibility into two and compare them separately.
Having potentials in all victory routes:
These are the civs that have a chance to focus on any one/two route(s).
Civs on the list: Germany, Australia, Rome, Aztec
My grading of the civs in this category:
1st: Rome, Aztec
2nd: Germany
3rd: Aussie
-Germany and Rome have greater potential if they have the opportunity to ripe, while Rome has a greater land mass to build districts and Germany has a production benefits that will be op in later game.
-Germany is powerful, but they cannot go religious, so this makes them less flexible in that sense, although they do really well on the other routes.
-Aztec has a earlier potential to gain districts, and they needs constant conquest as well as Rome does.
-While Rome has a infrastructure bonus and more powerful UU, Aztec has an amenity and strength bonus. Their strengths are similar and similarly powerful.
-Australia's potential is situational, and comes quite late as well.
Having advantage on various paths:
These are the civs that have a unique that covers multiple routes, but not doing particularly well on each of them.
Civs on the list: Japan, Brazil
My grading of the civs in this category:
1st: Japan
2nd: Brazil
-Both civs have advantage on every paths, but not too powerful.
-Japan builds districts faster and earn them in a faster pace from the beginning, on top of the adjacency bonuses.
-Brazil may catch up in earning Great People in later time, but it cannot pull up districts as fast as Japan.
Rounding up them all, in terms of dealing with situations (e.g. being DOW, having amenity problem, being surpassed in any of the victory routes), Rome and Aztec should stand out. Both possess strong military power and ways to support other route on top of the army. Aussie only does well in DOW, rendering their strength quite passive. In Japan and Brazil's cases, they do not have a convenient way to solve the problem.