Out of curiosity: what is Trip bringing to the table here that a regular attack or Precise attack wouldn't have? Alchemist's Fire is a ranged attack so Proning the target seems counterproductive. What am I missing?
Hit an upright target with AF and Trip Attack and suddenly they're prone (giving advantage to others who melee attack it) and on fire (taking ongoing damage unless they take an action to end it). Precise attack makes it more likely to hit, but it doesn't have the same "debuff their defenses" effect. The idea here would be that you could debuff the defenses and add ongoing damage to
four targets in a round if you spent all your superiority on it (mimicking a daily spike), while 4e-style
Crack the Shell is limited to one target, showing how it's not exactly potent in 5e's context.
Of course, you could also Precise Attack an AF and then also Trip Attack. Or a million other things. Battlemasters are able to respond to their context unlike any other class!
Hussar said:
But, there is a bit of a problem. All we're doing in more stuff in combat. Which is cool and all and does resolve a lot of the issues with a fighter constantly spamming the same two or three actions over and over again.
IMXP, the thing that saves the fighter from basic-attack-spamming is
fast combats. "Over and over again" doesn't apply very much when a fight lasts three rounds. That + terrain/cover/other tactical consideratoins + flexible improvisation...you might attack, attack, attack, for three rounds, but that doesn't mean you're not making interesting choices, including "Do I action surge?" or "Should I do an off-hand attack or burn my Seocnd Wind?" or "Do I huck an axe or close to melee?"
There should be interesting decisions every round, but more often than not, you are going to choose to attack and deal damage to your enemy - that's just how you win D&D fights.

Mages are doing basically the same thing in a fight, only their bows deal fire damage, or their caltrops are acid or whatever.
Hussar said:
How can this be translated to non-combat elements? How can I trade out attacks in order to inspire someone to do something? How do I trade out attacks in order to use skills?
You shouldn't need to - 4e didn't let you give up your level 5 daily attack to use a skill, either. These things are distinct pillars, and a character should baseline contribute to all of them without having to trade one for the other.
In 5e, if you have to make the choice between an attack and a skill, you might take the Dash action and make an Athletics check to leap over a cliff in a fight. Or if you want to inspire some NPC in a fight, you can make a Persuade check and say a few words (and a generous DM might even let you do that without having to spend an action, since "inspire" is pretty vague as to what the PC will then do). You won't always be doing these in combat, so you won't always be giving up attacks to do it.
More broadly, if you'd like to give up attack power to get more inspiring and more exploring, you're just a Fighter who invests in particular skills. Heck, your bog-standard Champion Soldier gets Athletics and Intimidation, and can take Insight and Survival, and if you go straight Strength/Con with your ability scores, with maybe Wis being 3rd, you are the go-to person in the party to climb that wall or jump that cliff, you take no BS, you can rough it for months without a problem, and you can inspire people the same way military commanders do: by making them afraid of the consequences of NOT doing what you want.

And that's without giving up ANY real combat power (well, maybe your ranged attacks are no great shakes, but you suffer no penalties for making Int your dump stat!). If you want to pump Charisma and be a Noble or be a Criminal and make a ranged fighter, or whatever, that's on the plate, too.