I've never played Takeda, so I can't comment accurately on how difficult they are - but I can say that in 100% of the games I play, they are the largest and most powerful faction in the early game. Unless I come into direct conflict with them at some point, leaving them to their own devices always makes them the most powerful NPC faction. This undoubtedly has a lot to do with their cavalry bonuses. A charge from a lot of horses has always been very powerful (in history and in these games). Regardless if they run into spears, a bunch of horses will cause just about any unit to break relatively quickly, and that's often all it takes to win a battle through chain morale failure.
Light Cavalry are powerful in their own right - for chasing down units and flank-charging the enemy. Being able to do it from the front with yari cavalry, and their added bonuses to attack and defense, along with their comparable mobility to light cavalry, makes them pretty damn good on the whole. Not to mention that there are more men in a samurai cavalry unit than light cavalry... AND they can dismount, so they can easily fill multiple roles in addition to being quick and sturdy.
I've honestly never thought that the cavalry were weak in this game. I rarely build them because they require more buildings than infantry, but I will always have at least 3-4 units of light cavalry (or just cavalry in general- 2 generals and 2 light cavalry or some other combo) for what they bring to the table. They can't hold up in a straight fight to infantry, being outnumbered and all that, but they're definitely not useless or underpowered.
And yeah, the start disadvantages to the Oda are well than made up for if you add in their bonuses to Ashigaru. Frankly, I've always thought Oda was one of the easier factions because of that. With that lowered cost, you can field larger armies and leave larger garrisons in towns for defense. I won't make full samurai armies until very late in the game, as I like to have a nice large unit of ashigaru handy for quick reinforcement. They really aren't bad units when they have enough experience.
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