You're Asking The Wrong Question
So my question is this: Why do you have a Web site?
Yes, Mr. or Ms. Advertiser, you have some recipes, maybe some coupons, games and/or fun facts, plus maybe an email list, and some corporate info, but you are underutilizing tracking and optimization, and you generally don't seem concerned with enriching the site experience. So I ask again, why do you have a Web site? It's always good to start with the fundamental purpose of the site, as it is a central component of a brand's digital footprint and the health of the overall business. From here it gets easier to understand the role search plays in your media mix.
In my opinion, if you buy TV, run any print, radio, and/or outdoor, then you'd better be buying search in proportion to the uptick in query volume. Search engines are where your consumers (B2B or B2C) are turning for more information when they begin to immerse themselves in a brand experience. Enable them -- don't make it hard.
Most advertisers already have a Twitter handle, a Facebook page, a myriad of high-profile high-priced endemic sponsorships, plus corporate information that can be easily indexed within local listings and directors anyway. So the next important question should be specific to where you should direct relevant searchers.
Many advertisers have a lot of media investments tied up in seeding, sponsoring, and owning key content relevant to their brand, yet we send most searchers to the corporate site. If a brand already owns health/finance/sports/coupon/recipe information on vertically branded sites, I'm going to go out on a limb here and say, use search to extend your media investments by leveraging your publishing and social community partners, their brands, and their existing SEO efforts to enhance your brand visibility within relevant SERPs. Drive traffic to these locations instead of sending all your traffic back to your corporate site, because this content likely has greater credibility and existing equity with visitors. Why fight these publishers in search and try to divert traffic back to your brand site?
The really big question then becomes how much search is appropriate. You might have to fund some research to get at this insight, as search doesn't have a ton of syndicated data available, but you can get at this answer. Also, don't be afraid to sink resources into your own SEO because of the inherent benefit it will have on your sponsored search efforts. For one, video SEO is a great way to get some extra mileage out of your creative and content investments as well.
Then the last key questions should be about technology and analytics. Whatever the answers are above and whatever you choose to do, it must incorporate a tracking solution so you can quantify performance and continually refine your efforts.
Are you going to run some modeling exercises and consumer research to feed your campaign criteria? Are you going to test some addressable TV against panel data to get some granular insight into the impact on search behavior? Do you plan to implement a tag management system to increase the accuracy and efficiency of managing the data collection? Do you have a system that can handle multiple data feeds so you can efficiently run the analysis and feed optimizations in near-real time? Do you have the capacity to feed/pull from CRM, store sales, etc. for retargeting?
This may sound daunting and it can take some time to sort out, but if you start with the proper questions, you can begin to find the right answers to build a game plan for better optimizing your media mix and ultimately enhancing your overall brand experience with search. The reality will be a combination of the right monitoring and tracking tools with a mix of advanced campaign management solutions, Web analytics platforms, simple site surveys, targeting, testing, and auditing tools.
The right questions and iterative testing will get you to what the right amount of search is for your brands -- not whether you should or shouldn't do search, because of course you should.