Cultural variants spread through cultural transmission, which can take many forms. Despite this, neutral models have provided an influential baseline for understanding transmission and innovation. But translating deviations from neutral expectations into specific mechanisms of cultural transmission requires quantitative predictions from models that depart from neutrality in controlled ways. Here, we develop age-structured extensions of neutral theory to explain systematic deviations in baby name distributions across multiple populations. We analyse two complementary perspectives on name diversity: variant abundance distributions (VAD, census data) and progeny distributions (PD, newborn names over time). Standard neutral theory predicts VAD power-law exponents of -1.0, but empirical data show exponents ≈ -1.5 with elevated fractions of both rare and common names. We demonstrate that age-constrained transmission, where only recently transmitted instances can serve as role models, generates this pattern through accumulation of culturally inactive individuals. Combining age constraints with anti-novelty bias, i.e., a preference for established variants, quantitatively reproduces the 1930 US census name distribution. For progeny distributions, we develop an effective neutral theory for thresholded data and show that datasets from eight regions are well-described by this framework after removing rare variants. Comparing fitted innovation rates reveals a new scaling relationship: larger populations exhibit systematically lower per-capita innovation rates. As this relationship is inconsistent with a mutation-like innovation process, which is agnostic to population size, we propose that this pattern reflects functional constraints on name choice. A key function of names is to distinguish individuals. But large populations do not require individuals to distinguish every other person — it is sufficient if distinguishability is ensured at the local scale. We operationalise this expectation as an anti-dominance bias, limiting the spread of any single name within local networks, which is able to replicate the inverse relationship between population size and effective innovation rate. Our results demonstrate how demographic structure, transmission biases, and functional constraints interact to shape diversity patterns.