Cine cardiac MRI is routinely acquired for the assessment of cardiac health, but the imaging process is slow and typically requires several breath-holds to acquire sufficient k-space profiles to ensure good image quality. Several undersampling-based reconstruction techniques have been proposed during the last decades to speed up cine cardiac MRI acquisition. However, the undersampling factor is commonly fixed to conservative values before acquisition to ensure diagnostic image quality, potentially leading to unnecessarily long scan times. In this paper, we propose an end-to-end quality-aware cine short-axis cardiac MRI framework that combines image acquisition and reconstruction with downstream tasks such as segmentation, volume curve analysis and estimation of cardiac functional parameters. The goal is to reduce scan time by acquiring only a fraction of k-space data to enable the reconstruction of images that can pass quality control checks and produce reliable estimates of cardiac functional parameters. The framework consists of a deep learning model for the reconstruction of 2D+t cardiac cine MRI images from undersampled data, an image quality-control step to detect good quality reconstructions, followed by a deep learning model for bi-ventricular segmentation, a quality-control step to detect good quality segmentations and automated calculation of cardiac functional parameters. To demonstrate the feasibility of the proposed approach, we perform simulations using a cohort of selected participants from the UK Biobank (n=270), 200 healthy subjects and 70 patients with cardiomyopathies. Our results show that we can produce quality-controlled images in a scan time reduced from 12 to 4 seconds per slice, enabling reliable estimates of cardiac functional parameters such as ejection fraction within 5% mean absolute error.