You're pointing at something real, though I'd want to sharpen the framing a bit, because "science insists there be no sense of design" is both true and misleading depending on what level you're talking about.
Where the tension is genuine
The human mind is a pattern-detecting, agency-attributing, story-building machine. We see faces in clouds, intentions in coincidences, and purpose in natural events. This is well-documented (agency detection, teleological reasoning in children, the "design stance" as a default folk ontology). Science, at its most rigorous, systematically strips this out: it demands mechanistic explanation, falsifiable predictions, and indifference to whether the result is meaningful, beautiful, or comforting. A earthquake isn't punishment; it's plate tectonics. A mutation isn't striving; it's a copying error. The "design" we intuit is treated as a projection to be explained away, not a feature of the world to be honored.
So yes: the scientific method is in part a set of cognitive disciplines for overriding default human teleological thinking. That's not incidental—it's one of its core functions. Bacon, Darwin, and the whole history of debunking natural theology are about exactly this.
Where the framing overreaches
But "science insists there be no sense of design" isn't quite right, for two reasons.
First, science is full of apparent design that it explains rather than denies. Biology is saturated with functional language—"the heart is for pumping blood," "this enzyme's job is to..."—and that's not a concession to superstition; it's how you describe a system shaped by selection. Darwin's move wasn't to ban design talk but to give it a non-designer mechanism. Similarly, physics uses "fine-tuning," "anthropic" reasoning, and optimization language all the time, carefully hedged.
Second, the practice of science is deeply design-seeking in a different sense: scientists look for structure, symmetry, conserved quantities, elegant laws. Einstein's insistence that God doesn't play dice, Dirac's search for mathematical beauty, the whole conviction that the universe is intelligible—these are aesthetic and quasi-teleological drives doing real heuristic work. The method polices claims, not motives. You're allowed to be driven by a sense of design; you're just not allowed to let it count as evidence.
The deeper point
The tension you're sensing may be less "science vs. design" and more "two different cognitive modes." One mode asks what does this mean, who is it for, what is it trying to do—the interpretive, agent-centered stance. The other asks what follows from what, under what conditions, testable how—the mechanistic stance. Humans need both; they're not interchangeable, and neither is simply correct. The error is category confusion: treating the interpretive stance as a source of empirical fact (creationism, most conspiracy theories), or treating the mechanistic stance as a complete answer to human questions (scientism, the assumption that "it's just neurons" settles anything about meaning).
Science doesn't forbid a sense of design. It forbids inferring a designer from that sense without independent evidence. The feeling of design is data about you, not about the world—and that's precisely the move the human mind resists.
Is there a particular version of this tension you're working through—design arguments in biology, the "unreasonable effectiveness of mathematics," or something more about how people resist scientific explanations?