These are major problems with the wikis and here are some ideas on how to improve this.
Firstly, why at all?
Tag Wikis are an awesome resource. Stack sites' have a lot of expertise and wikis are one way of documenting and showing it to new users. Tag Wikis can contain some of the canonical information as well as stuff that would not normally be a good fit for the Q&A format, even though the expertise is there, and so is the willingness to help.
Making Tag Wikis searchable should not / would not be a very difficult task. They are indexed by Google, so they are searchable externally.
I think they should be made searchable similar to questions and answers internally as well. In a search query, the tags should first be identified, and then the remaining terms could be searched for within the wiki. Then, questions and answers could be searched normally. If the Tag Wikis are well written and curated, they would have the FAQs of the tag and some of the commonly asked information as well, making the search process possibly faster and return relevant results.
Tag Wikis are a minimum 3 clicks away* from the home page whereas questions and answers are 1 click away. This itself is a demotivating factor as far as putting efforts into the wikis are concerned as compared to asking and answering. Moreover, the chances of a new user even stumbling onto the wikis is remote.
To tackle this, Tag Wikis could be brought right onto the home page. Clicking on a tag from the panel on the home page would redirect to the wiki and not to the tagged questions. This would be only from the top panel.
The tags that appear on the top panel would be ones that are selected via a democratic process by the community, and not simply the most popular ones. This would allow for the site's best wikis to be displayed on the home page.
Too much space?
This would occupy a lot of real estate and the reason why I think some text should be kept (contrary to the current implementation) is so that new users see it, and then may get curious enough to click through. If we simply had tags, it would not highlight the wiki as prominently, thereby reducing the chances of even getting mouse-overs. It could however, be made dismissable, or not appear for users having rep>some limit as they most likely would know about the wikis.
Depending on the implementation, most of the high volume established sites would not want this IMO, so this request is targeted towards the newer SE sites, especially beta sites. Beta sites have a much lower volume of questions but not any lack of expertise. Hence, putting efforts into resources such as the wiki is viable for newer sites. It can become an excellent way of showing and documenting the community expertise for new users.
*Wikis are reachable by hovering and clicking on info. However, considering there are 6 clickable options on the hover-card plus hovering by itself is not something new users would do, this is rather un-intuitive.