What makes a cooperative board game actually a game?
What is a game?
There's some argument about what exactly constitutes a &quot;game&quot; in academic and design circles. Going by Salen and Zimmerman's definition from Rules of Play:

A game is a system in which players engage in an artificial conflict,
defined by rules, that results in a quantifiable outcome.

Cooperative games still have rules and definite win/loss outcomes. &quot;Artificial conflict&quot; does not prescribe that the conflict be between players, or that it be focused on strategic competition per se.
Strategy vs. algorithm
The operative assumption in the original question is that play vs. a sentient actor requires on-the-fly decision-making, whereas cooperative play vs. random game events can necessarily be reduced to a script you can follow to win. I think this is an error.
Even when the adversity they face in the game is tied to something as simple and unthinking as the luck of the draw, the human players may still have to adjust their actions to the unpredictably changing game state. To use your Dominion example, simply knowing what an optimal strategy looks like in the abstract doesn't free you from having to adjust your turn-by-turn plays to make the best use of the cards you've actually drawn.
You could try to articulate all these tricks into a single defined &quot;strategy,&quot; but for many games this hypothetical script will be so complex that a human player couldn't simply follow it by rote. And how is this different from trying to write out a gameplan that includes ways to compensate for intelligent opponents' moves, anyway?
Counterexamples: strategic depth without opponents
Consider these examples of challenging decision-making without a human opponent:

Classic solitaire games such as Klondike involve hidden information and strategic decisions analogous to classic multi-player card games.
Knizia's Lord of the Rings board game is a cooperative board game with a random element (some tiles in a bag) as the the source of adversity. It has an expansion that allows a player to assume the role of the antagonist; this tends to increase the difficulty a bit since the Sauron player is actively choosing the most punishing outcome, but either mode involves qualitatively similar gameplay and strategic decision-making for the ringbearer players.
There's a casual variant of Magic: the Gathering where players team up against a random deck representing a zombie horde. The horde just immediately casts whatever it topdecks and always attacks with all of its guys. There are certain things the horde can't do (play targeted removal, for instance), but the players still have to make strategic decisions about resource utilization, attack vs. defense, and timing their spells that are qualitatively similar to the decisions involved in a normal game of Magic. This is a mirror of the situation above: a normally head-to-head game can be played as a cooperative game against a random element, with qualitatively similar gameplay and strategic decision-making.
Video game AI opponents are often actually fairly simple actors just following a generally-good strategy. Despite not being particularly reactive to the actions of the human players, these rather simple opponents can still present a significant challenge to them.

Is strategic thinking an integral part of all board games?
More generally, I think your question presupposes too much about games and why people play them. Board games, like all games, can provide a variety of experiences, including:

Overcoming a challenge
Proving yourself against your peers
Solving a puzzle
The thrill of gambling
The visual and tactile pleasure of gameplay (think about how much enjoyment miniatures gamers get from the look and feel of their pieces)
Socializing with your friends
Experiencing the narrative of play (drama and resolution, for instance)

Among these, overcoming challenges is not always and necessarily the game's or players' primary goal. Nor is strategic play the only way to experience a challenge: games can easily involve physical and mental tests that don't depend on analysis and planning.
Some products traditionally labeled &quot;board games&quot; don't actually put much emphasis on overcoming challenges. Candyland is, of course, the archetypical extreme example. Some game designers and purists may say that such board games aren't really &quot;games,&quot; but that is their popularly-accepted label.