We introduce a set of tools which simplify and streamline the proofs of limit
theorems concerning near-critical particles in branching random walks under
optimal assumptions. We exemplify our method by giving another proof of the
Seneta-Heyde norming for the critical additive martingale, initially due to
A\"id\'ekon and Shi. The method involves in particular the replacement of
(truncated) second moments by truncated first moments, and the replacement of
ballot-type theorems for random walks by estimates coming from an explicit
expression for the potential kernel of random walks killed below the origin. Of
independent interest might be a short, self-contained proof of this expression,
as well as a criterion for convergence in probability of non-negative random
variables in terms of conditional Laplace transforms.