How should I approach companies about sponsoring my TV segments to get them interested?

I produced a show for three seasons with a couple techie friends. Getting paid sponsorship usually requires numbers: demographics, viewership.

The person making the sponsorship decision wants to know that the viewership is in sufficient quantity and the right age range, socio-economic status, etc. to be a fit with their own target market.

You can calculate an ROI with these numbers--an agreed-upon # of viewers = so many active prospects for the company = a certain # of buyers at $price = total revenue. But it's an educated guess until you have a track record.

Sometimes, especially if it was a marketing agency that ran all these decisions for the end customer, I was told if they went ahead it would be because of "goodwill" and not an ROI-based choice.

Local programming did matter. Demonstrating you could get at their target market was key.

That sounds interesting, may be something I am interested myself or can help directly. With that said, I don't want to spam or pitch, so let's see...
Based on my experience meeting with clients and having done some projects and appeared on television shows I can say that a lot of the advertising business developers work mainly off networking in person. Relationship building for pipeline building. Obviously, as marketer, I think there are better more efficient ways but networking, in my opinion, will always be the cheapest and easiest and in some cases the most effective.
What you need is a funnel system with case studies and examples serving as ads for your own end goals which is to sign up brands. Partner with potential vertical companies that can help you sell spots more regularly such as established ad agencies and magazines.

You need a good process in place for the outreach. Cold email and quality lead generation is the way to go.

I would start off with building a high-quality lead list with correct decision maker's contact info. Then I would move onto an outreach process that starts with cold email and create multiple touchpoints including phone calls to maximize results.

Have a solid strategy in place for finding sponsors within your niche.