Day 1: enter the Gothic Quarter first thing, then Cathedral–Plaça del Rei–Sant Jaume; touch the Rambla only in passing; cross to El Born to eat and wander; evening near Born or nearby waterfront.
Day 2: Sagrada Família early, then Sant Pau and Eixample / Passeig de Gràcia — timed icons in one geography; solid dinner away from being glued to one monument.
Day 3: Park Güell (booked), Gràcia for lunch, Montjuïc by default — or swap Montjuïc for Barceloneta and the coastal walk for a sea close.