Planetesimal formation via the streaming instability of small pebbles in protoplanetary discs requires enhanced solid concentrations relative to the solar metallicity, Z≃ 0.01. We investigate here whether pile-ups in weak axisymmetric pressure bumps are sufficient to trigger strong particle concentrations via the streaming instability. Using high-resolution 2D shearing box simulations with millimetre-to-centimetre pebbles, we explore the behaviour of the streaming instability in the presence of a non-reinforced pressure bump. We find that even very weak bumps, with gas density amplitudes as low as A = 0.04 relative to the background, produce dense particle filaments via the streaming instability for all tested Stokes numbers at solar metallicity in the inner disc. Outer disc regions require slightly stronger bumps (A ≥ 0.14) to form filaments at a solar metallicity, though the necessary bump amplitude is substantially lowered when the metallicity is increased to Z = 0.02. Our results suggest that weak, non-reinforced pressure bumps can act as focal points for planetesimal formation via the streaming instability in small pebbles, in contrast to previous studies which used reinforcement to maintain the pressure bump. Additionally, we introduce a pressure-bump-dependent clumping criterion, (Z/χ)crit ≈ 0.3, where χ = Πmin²/Π₀ reduces to the background pressure gradient Π₀ in the absence of a pressure bump. This criterion encapsulates the scale of solid pile-ups at lower pressure gradients and accurately predicts the onset of strong clumping based on the results of our simulations. With pressure bump signatures being common features of observed young discs and magnetohydrodynamical simulations, the results of our 2D simulations imply that weak pressure bumps may be major cradles for planetesimal formation in protoplanetary discs.