I will give one good argument for your 'should'. Quoting from the last Nature Reviews paper in the list above:
Research in P.J.M.'s laboratory has been supported over the years by the Swiss National Science Foundation, King Abdullah University of Science and Technology (KAUST; Saudi Arabia), the University of Lausanne (UNIL; Switzerland), École Polytechnique Fédérale de Lausanne (EPFL; Switzerland), Centre Hospitalier Universitaire Vaudois (CHUV; Switzerland), the National Centre for Competence in Research (NCCR) Synapsy and the Préfargier Foundation
Clearly Nature is paywalling, at least partially, publicly funded research results.