I'd say part of the source of your problems is using yari samurai in a role they are not well suited to. They are too easily countered by units like katana samurai, lack heavy armour, have a small unit size, and can't use spear wall. They are a reserve unit and anti-cavalry unit, intended to use their speed to rush to hotspots on the field. In the main line they don't fare so well. I never field more than one in an army. Katana samurai are a far better choice for killing power, and naginata for survival and versatility.

I don't think the legendary AI gets a bonus to reload. I've not heard of it before and don't remember seeing that when I played on legendary. I think you're being scuppered by your Daikyu - they have the attribute "slow reload speed". They are more of a sniper unit than a front line archer.

Under similar circumstances I and a few others adopt a keep it simple approach:

I'll use 8 archer units. They will all be of the same type so that range, capability etc is identical. Thus, either bow ashigaru or bow samurai. Bow monks are too expensive and don't have the durability for this role. I'll then have 8 strong infantry units, either yari ashigaru or naginata. These units are intended to hold, not die, and do some killing. Survivability is the key factor for them. With a general in the army this leaves space for 3 units. These are a bit open to choice. I'd go with heavy cavalry, or samurai with high killing ability like katanas or no dachi. If you want you can try using a single sniper unit like the monk archers or daikyu, although personally I find that makes my army feel a bit too lacking in hand-to-hand potential. I would never use gunpowder in this bow heavy army; it's not a good synergy as it demands a lot of micromanagement right at the point where I already have my hands full bringing my infantry line into play. Units like yari samurai are great as reserves in other army styles but I don't like them in this one as they lack the ability to inflict heavy casualties quickly. Remember: fancy or uncommon units replenish more slowly on the campaign map. This can lead to your army being held up for a long time while it recovers after a tough battle, and on the highest difficulties that's not something you want.

I'll aim to recruit the bows in a province which gives bonuses to accuracy and armour, with accuracy being the most important. The infantry and cavalry get attack and armour bonuses. Research the arts which give your archers a bonus ASAP, particularly extra ammo and fire arrows. On the infantry side, research arts which boost up your naginata. Focused, deep research is worth more than wide, low-level research when using this army as you are only fielding a few unit types.

Because there are exactly 8 units of bows and infantry, I'll deploy in two parallel lines. The bows go in front in a single line set to loose formation. The infantry in a single line behind them, a little way back but not too far. I'll look for terrain which supports this by protecting one or both flanks. If there's no useful terrain it's not a total disaster provided I keep my eyes open for flanking attempts by the AI. If it's a defensive battle where I can deploy screens I will do so. I'll shoot great clouds of arrows and fire arrows at the approaching enemy, and when they get close to contacting my bows I'll either pull my bows back behind my infantry line or run my infantry on forward through my bows to engage. I tend to advance infantry when using naginata and withdraw bows when ashigaru and their spearwall. If all has gone well you have a bunch of half dead, scared units engaging your infantry line. They will hold and start doing some damage. Now you need to ensure all of your bows have good positions to keep shooting, and to start using your other units to flank the enemy and begin dicing. From there it should be a simple matter of directing your melee units towards new targets as enemy units begin to break.

Unless there is a specific target I want to shoot I let my archers pick their own targets by enabling 'Fire at will'. This reduces micromanagement considerably, and generally they will target intelligently.

If the AI has a large number of bows and sends them forward ahead of its infantry I'll order my archers to shoot the enemy bows, focusing fire so that several of my units shoot at one of his. The AI doesn't like to use loose formation so my archers will do more damage than his unless there's another factor in play like superior unit experience. When the target is down to around 50% strength I'll target the next healthy bow unit. The aim is to weaken the bows and render them fairly safe to ignore until a more opportune time. Destroying them in an archery duel takes a lot longer and uses many arrows, and tends to leave me too weak to inflict sufficient damage on the enemy infantry. Alternatively, I'll let my bow units play with his and run my infantry forward to aggressively attack the enemy army. In this scenario I want my bows to be taking most of the incoming damage so that my infantry arrives mostly intact.

On higher difficulties it's entirely normal to have costly victories and to lose battles. There's no way to avoid this as the AI in Shogun 2 is far better than in the older games.