The introduction of pre-trained transformer-based contextualized word embeddings has led to considerable improvements in the accuracy of graph-based parsers for frameworks such as Universal Dependencies (UD). However, previous works differ in various dimensions, including their choice of pre-trained language models and whether they use LSTM layers. With the aims of disentangling the effects of these choices and identifying a simple yet widely applicable architecture, we introduce STEPS, a new modular graph-based dependency parser. Using STEPS, we perform a series of analyses on the UD corpora of a diverse set of languages. We find that the choice of pre-trained embeddings has by far the greatest impact on parser performance and identify XLM-R as a robust choice across the languages in our study. Adding LSTM layers provides no benefits when using transformer-based embeddings. A multi-task training setup outputting additional UD features may contort results. Taking these insights together, we propose a simple but widely applicable parser architecture and configuration, achieving new state-of-the-art results (in terms of LAS) for 10 out of 12 diverse languages.