Google search is meant for people to find answers, products, services and other information.
Why would Google place a single page website in the SERP for a query besides the name of the business?
What questions does it answer?
What value does it provide?
For a hospital you probably want to focus on local search.
Do they have Google My Business configured and optimized?
Are they added to all local business directories?
Best you can probably do is show up in search based on proximity. If the client doesn’t want to invest in content then they don’t want to invest in SEO. SEO requires content. If someone asked me to do SEO but said they don’t want any content I would not work with them.
At minimum you need service pages that explain each service. Even if you run ads you’ll need specific landing pages for each type of patient you will want to attract. You can’t just send people to a home page from an ad and expect that to convert at a level that would be profitable. Backlinks aren’t going to help you and the likelihood you could get very many quality backlinks is low.