Unlike shared server hosting, where there's little work involved in setting up a basic account, and it can be done in a few minutes, setting up a dedicated server is more time consuming. It also dedicates an entire server, so the cost of hardware must be considered. For those and other reasons, I think it highly unlikely you'll find a free trial period.
However, the next best thing is available. Some companies do have a 30 day money back guarantee.
I think free trials would be a lousy idea for any host. They would have a lot of work to do for nothing. People unhappy with their present host would try out a dedicated server to see what it's like. Many would find out why they aren't qualified to manage one, and they would drain the resources (including tech support) of the company in the process. That means that paying customers would suffer.
Years ago, many software comapnies used to offer unlimited free support. At its heyday, Word Perfect even had toll free lines for support. Eventually, they all dropped the unlimited free support, even WP. They had to.
The simple reason is that too many people took the easiest path to getting an answer, Instead of opening the manuals, they would pick up the phone to ask simple questions that might have more to do with the operating system than the program. Meanwhile, customers who had a real problem, had to wait much too long while tech support dealt with the lazy.
That's why you see limits on support now. Support may be free for a limited time, like 90 days, and is not likely to be unlimited, at least not with any popular program.