It's Autocracy though that appears quite weak. Don't most war monglers find they've already effectively won before the Industrial era even starts?
That used to be the case, but I've found it to be much more viable to start warmongering late in BNW. Late game wars are very easy and quick, especially when you have the warmongering might of Autocracy.
On that note, though, Autocracy isn't just for warmongers, and in fact, if you find yourself in a large lead late game, it enables you to leverage that huge army in other ways. It also allows you to catch up from a tech deficit far more easily than the others, if you find yourself behind. It basically makes your closest CS allies impenetrable to outside influence, as well as giving you a massive, stackable boost to influence in Cult of Personality.
Freedom's culture T3 is arguably the weakest, because even though it is the most versatile, Freedom has no big "everything" modifier to really slam it out against differing ideologies. Order's Culture T3 is potentially the strongest, but only on a narrow band. Differing ideologies once again give it trouble, but not nearly as much as Freedom, since both of its boosters are "everything" modifiers. Autocracy's is without a doubt the strongest, though it requires an active approach. It can't be used passively at all, but it gets a BIG +50% "everything" modifier that can stack with itself when at war with two common enemies, leading to staggeringly easy Influential status. Freedom can more easily target diplomatic bonuses with its Diplo T3, but it is once again weaker than the Autocratic alternative, and is completely useless during wartime, since enemies can sink its trade ships, and even without a war, freedom's Gold income must suffer to use it. Freedom can more easily purchase spaceship parts, but it gets no direct bonuses to science, while Order gets several, which means you need to stay small and really leverage your specialists in order to win the space race -- if you get too big, or if you get too many puppets, that science penalty will rear its ugly head, and buying spaceship parts won't matter, since Order can get the science first.
So, basically, this is how the three ideologies work:
Freedom has the most versatile, easy-to-use peacetime bonuses, and heavily leverages tall empires, but numerically has the weakest T3 in each of its classes. It plays the best in isolation, but is unlikely to catch up if Order and Autocracy are actively working against it, since having to stop what it's doing to build an army that it actually has to USE prevents it from doing what it wants: to sit there and build. Many of its advantages have a hidden drawback, and while it can certainly build an army, go to war, have puppets, and train its military, it has a much higher opportunity cost for doing so.
Order leverages wide empires with its numerous per-city bonuses, and it benefits when a majority of the other civs pick it. It is also the best at keeping others from meddling in its affairs -- once Order gets a lead, it keeps its lead. Enemy spies get machinegunned to death, its production is huge, its science gets direct injections, and its influence with other Order civs is unmatched, making it unlikely for ideology unhappiness to affect them. Unfortunately, Order has no means of influencing City-States, so it often loses out on alliance bonuses as Freedom and Autocracy gobble them up, and as a result, Order will end up with little influence in the world congress. Order NEEDS to get a lead to take advantage of its tenets, or else it will stagnate. Without a lead, Freedom gets to leverage its money and city-states, Autocracy will steamroll right over them with powerful armies, and other Order civs will use the weaker ones as stepping stones. In fact, if you aren't winning as Order, you'll most likely lose to someone that IS winning as Order.
Autocracy has the largest bonuses in every category it's in, but virtually none of them are passive, aside from the happiness it gets, which are insane. Autocracy requires active commitment and a deliberate plan to take full advantage of its power. It is without a doubt the best ideology for a player who fell behind and needs to catch up, but it does not hold onto a lead well at all, unless it constantly smashes down the others who might try to usurp its mantle. Fortunately, everything it does leverages having a huge army, making that army useful for important things other than conquest, but if you procrastinate or try to get comfortable just sitting there building, you lose.