I wanted to share a solution in case anyone runs into the same issue. The problem stemmed from Notion using shared workers to improve performance (you can read more about this here: https://www.notion.com/blog/how-we-sped-up-notion-in-the-browser-with-wasm-sqlite).
This caused Playwright to crash, leaving the process stuck.
To resolve it, I added the following line to the Docker Compose environment:
DEFAULT_LAUNCH_ARGS=["--disable-shared-workers"]
This disabled the shared workers feature when launching the browserless, and that fixed the issue.