Most important trade to know to open a home remodeling business?

Karate.

No need for a saw.

I think the most important trade is experience in this field. I have been a re-modeler/home builder for several years and have been through all of the training and the courses and yes that is good knowledge but hands on in this trade is a very important part.

I have been doing this for over 20 years and I am still learning.

Management.

If you have the ability to manage people and money you don't need a specific trade.

Anyone can start any kind of business. Those who prosper are those who either learn something about the work itself and also have that 6th sense about what people expect of your business.
To answer your question, Carpentry is the primary trade needed for remodeling. I started out as a working carpenter/owner. After 5-6 years our work and time to do estimating started reducing my time with the tools. After 10 years I was limited to being on the job about 50% of the time. Today I hardly get a chance to strap on the toolbelt.
Carpentry experience is very valuable. You can go to seminars, and buy the estimating books,software. You can make the mistake of trying to grow too quickly.
Your attitude in dealing with customers is essential. That is why some of the best carpenters could not survive if they ran their own business, they are better off working for someone, but if you have good carpenters working for you, you also need to be able to step in and work with them if things get really busy. When you do give them a hand there is nothing better for them to understand than - Hey this guy knows what he is talking about.