While building a rank, say for a search engine, or a recommendation system, is it valid to rely on click frequency to determine the relevance of an entry?
Depends on the user's intent, for starters.
Users normally only view the first set of links, which means that unless the link is viewable, it's not getting clicks; meaning you'd have to be positive those are the best links, otherwise the clicks are most likely going to reflect placement, not relevance. For example, here's a click and attention distribution heat-map for Google search results:
Further, using click frequency to account for relevance is not a direct measure of the resource's relevance. Also, using clicks is problematic, since issues like click-inflation, click-fraud, etc. will pop-up and are hard to counter.
That said, if you're interested in using user interaction to model relevance, I would suggest you attempt to measure post-click engagement, not how users respond to search results; see "YouTube's head of engineering speaking about clicks vs engagement" for more information, though note that the size itself of the content is a factor too.
Might be worth noting that historically Google was known for PageRank algorithm though it's possible your intent is only to review click-streams, so I won't delve Google ranking factors; if you are interested in the Google's approach, you might find a review of Google's Search Quality Rating Guidelines.
For my part I can say that I use click frequency on i.e. eCommerce products. When you combine it with the days of the year it can even bring you great suggestions.
i.e.: We have historical data from 1 year over 2 products (Snowboots, Sandalettes)
where  = January
As you can see, snowboots are much more searched in January than sandalettes, so you should suggest snowboots to someone searching shoes on your site or /we on january.
You can also see if something is "fresh" at this time, like when people often click a unknown product it could be an insight for a new comming trend or something.
That are just some examples where you could use click frequency as an insight. I think there are no rules for what you can use or not in recommendations, as long as it makes sense.
Is it valid to use click frequency, then yes. Is it valid to use only the click frequency, then probably no.
Search relevance is much more complicated than just one metric. There are entire books on the subject. Extending this answer beyond a simple yes/no would likely make the answer far too broad (and opinionated)