Infrastructure-as-Code (IaC) generation holds significant promise for automating cloud infrastructure provisioning. Recent advances in Large Language Models (LLMs) present a promising opportunity to democratize IaC development by generating deployable infrastructure templates from natural language descriptions. However, current evaluation focuses on syntactic correctness while ignoring deployability, the critical measure of the utility of IaC configuration files. Six state-of-the-art LLMs performed poorly on deployability, achieving only 20.8$\sim$30.2% deployment success rate on the first attempt. In this paper, we construct DPIaC-Eval, the first deployability-centric IaC template benchmark consisting of 153 real-world scenarios cross 58 unique services. Also, we propose an LLM-based deployability-centric framework, dubbed IaCGen, that uses iterative feedback mechanism encompassing format verification, syntax checking, and live deployment stages, thereby closely mirroring the real DevOps workflows. Results show that IaCGen can make 54.6$\sim$91.6% generated IaC templates from all evaluated models deployable in the first 10 iterations. Additionally, human-in-the-loop feedback that provide direct guidance for the deployability errors, can further boost the performance to over 90% passItr@25 on all evaluated LLMs. Furthermore, we explore the trustworthiness of the generated IaC templates on user intent alignment and security compliance. The poor performance (25.2% user requirement coverage and 8.4% security compliance rate) indicates a critical need for continued research in this domain.