Can a free-to-play player compete with someone who spends?

Yes, partly — especially on finesse. A spender goes faster and further in raw terms, but Dragonfire's combat rewards things money doesn't buy:

• The affinity triangle (Cavalry/Archers/Spearmen/Shieldbearers) and the +7% edge on the right target
• Scouting and the right matchup before you attack
• The synergies between an Order and its ⭐ set Habit, auras, control effects (Panic, Stun…)
• Choosing the star tiers that unlock an effect

Where F2P loses ground: speed (timers, dragon level, Relic volume) and certain instant jumps.

Neutral conclusion: you won't beat a big spender in a race for raw Power, but you can win fights against them by playing better, and progress solidly over the long run. Playing well stays free.