Every combat round relies on luck, Drill just gives a few extra rounds
There's a good combat mechanics article in the war academy, but to summarize here:
Assuming an even battle between two units, both start with 100HP, do 20 points of damage when they win a combat round, and have 50% chance of winning a combat round. That means that one has to win 5 out of 9 rounds. That's still 50% odds to win the combat even if 5 out of 9 looks better than that
To gain better odds in winning the combat you can try to affect the variables: win with fewer combat rounds required, have more combat rounds out of which you need to win, and have higher chance to win each combat round.
Having higher effective strength than the opposing unit does many things: you do more damage per round, opponent does less damage per round, you have higher chance to win each round. Having just 0.1 points higher effective strength results in opponent doing only 19 points of damage, which is quite significant: opponent needs to win 6 combat rounds instead of 5 while you still need only 5, therefore you need 5 out of 10 instead of 5 out of 9. You also have a slight advantage on each round (C1 warrior vs green warrior is about 52.4% chance on each combat round).
First strikes don't effect the damage done per round, nor the chance to win a combat round. They just add combat rounds out of which you need the same number to win. Having one first strike with equal strength means 5 out of 10 rounds - almost as good as having marginal strength advantage (from C1).
Drill is quite little used promotion because the first level grants only a first strike chance. Second grants first strike, and the going gets better with third and fourth, for a grand total of 3-6 first strikes. Of course for those you could've taken combat 4, which would be strong in itself. Except of course if you're protective and are promoting archery unit and so have that Drill1 to start with.. Comparison would be 3-5 first strikes (Drill4 compared to Drill1) vs. +30% base strength (C3).
The next step in affecting combat rounds required using brute strength is at roughly 40% strength advantage when the opponent needs not 6 but 7 combat rounds to win. It's a slaughter at that point, as we're talking about 60% odds per round. At 60% strength advantage you finally do 25HP per hit and thus need only 4 rounds to kill the sucker.
As you can see, the first big step is gaining the upper hand, however minimal. After that, first strikes shine. Or the other way to look at it: if you have bad odds to begin with, first strikes even them out.
The major point here is that there are a couple of steps in the combat odds. Affecting the chance to win a round in a big way is quite hard, requiring BIG strength advantage. But the unit holding even the smallest strength advantage immediatelly has an advantage in damage done per round and therefore rounds needed to win the combat. Note: rounds needed to win combat. The biggest advantage was actually exactly the same as that provided by first strike
Now there was this one more way to affect the combat odds: make sure your opponent is not in full health.. This grants multiple advantages, with the obvious one being that you only need to whack him a few times to take out the remaining breath, so again you need fewer combat rounds to win (this seems to be the theme). The other is that the effective strength of the unit is the average current and full health multiplied by normal strength. This further skews the odds in your favour. A full health longbow is tough, but take him down just a notch (from 6 to 5.4 strength) and the effect is huge.
Obviously first strikes are best when you already hold the upper hand (longbows defending city commonly have higher effective strength than the attacker, which is why cats/trebs are used - to soften the defense stack with collateral damage) as then you start with combat rounds that you have good chance to win while having no chance at all to take damage on.
The choice of promotion is almost always a hard one. Sometimes it's not very hard - giving garrison promotions to longbows that are not expected to ever leave a city is quite sound. As are raider proms for maces who should never attack or defend except when reaching a city (other units defend the stack and pick the opposing units on the fields). But outside the specific city garrison / city raider unit specializations, it's not that clear.
Straight combat prom is of course always an option. It's good in that it's very clear on what it does: +10% strength. It gets applied to the unit's base strength in all cases, with no situational factors. Any other strength-modifying proms are counted against your opponent, therefor their effetiveness depends on the base strength of your opponent. If the base strength of your opponent is low (compared to yours), and the opposing unit relies on high bonuses, a straight combat prom may be more effective than a counter prom. This is very rare though, as counter proms (which includes garrison and raider, but mostly means shock, cover, formation, pinch, as well as the anti-siege and anti-armor the names of which escape me now) are 25% while combat is 10% - you basically need 2.5 times the strength of the opposing unit for combat prom to be better (knight vs spear eg - against pike shock is better for the knight).
So, as the straight combat prom is unlikely to be as good as the counter prom, why take it? Because it works against all unit types.. Can I be sure this unit will only face some specific type of unit? Combat works against all opponents. Attack archery unit now, melee unit the next time - combat works in both cases while shock would be wasted in first, cover in the second. Counters work best when defending, as the defending unit is chosen against the attacking unit (best defender against the specific attacker is the one to fight), which deters the opponent from using eg. mounted units while you have formation pike in the stack, and melee units when you have shock crossbow in the stack. Which of course brings us back to cats which do collateral damage..
(for the record: I find it sad that the only way to deal with any stack is to suicide collateral damage on it. No tactics, no strategies, just massed cats used expendably)
Drill, like Combat, is not really that situational. It effects all combats regardless of the opponent. With the exception of first strike immunity, but considering that those units are already counters to the first striking units, being forced into that battle would be the same as being forced into attacking axe with a sword.. It happens at times - things are bad and you need to take emergency measures.
The problem I see with Drill is that it has a strong curve: Drill1 is not very useful (Drill2 OTOH is), Drill4 is immense compared to "just" Drill3. Therefore, commiting a unit to Drill promotion lines is a long term plan where you expect to get up to Drill4. However, similar to Combat proms, they unlock the counter proms which may be useful: shock crossbows are nice too.
The advantage is of course that the first strike rounds are fought first, and you can't take damage on those rounds. But for that advantage to be meaningful, you again need strength advantage over the opponent or MANY first strikes.. Otherwise it's just the same old luck.
