Sorry for the necromancy but I had a slant to add and too long for comment.
Viewpoint: It's not reasonable for the students to eschew previous techniques. But it's ALSO not reasonable for you to be so super surprised that they lack perfect recall of previous material. Instead of this all-or-nothing attitude, why not a more sympathetic approach: "yeah, I know partial fractions integration was a while ago and it was at the end of a long course...but we actually need that thing now...don't worry, we won't rehash every tough integration technique but this snake needs to come out of its hole again...no sweat, let's walk through the problem slowly and show the steps we use."
Course design (you asked): I'm not sure that proofs is the key thing to working on with a course where you are trying to give "exposure". For someone who uses a technique a proof is really more like a motivation for why you can use the technique. But the technique is the important thing, not the proof.
Course structure: I would also suggest designing the course to follow a simple text, not as an ad hoc set of subjects or something that requires following lecture notes or stitched together handouts. This gives the students some supporting scaffolding.
Pedagogy: The surprise at imperfect students and the attention to proofs makes me suspect someone who is good at understanding MATH but weak on understanding STUDENTS. The math is half the problem. But understanding students and how to improve them is PART of the problem. Yes, absolutely, it is! You can't ignore it, just as you can't ignore other independent variables in multivariable calculus, just as you can't ignore interactions and higher terms in stats problems that are nonlinear.
Course analysis: Take a look at your course and take a look at the previous calc course (or even algebra). Figure out the key concepts that usually trip students up and make a little cheat sheet or refresher formula list or even list of key sections in the actual text used most commonly in your uni (or in nearby high school). [This would be given to the students on day one.] But note, the point here is to be thoughtful and analytical and find the MINIMUM that they need to know. Force yourself to list what is needed and not needed. And then within the needed, list the things that typically confound students, not the ones that are usually no problem. So, instead of saying "you should know that whole calc book"....well, sure they SHOULD. But we are dealing with imperfect beings. Not angels. Not robots. So if they need integration by parts (and it usually is an issue) but they don't need partial fractions, list parts integration and don't list partial fractions. Figuring this out is part of your JOB. And note that it is a relative thing (very different from Euclean proofs). This is about max gain for min pain (AND with imperfect information). At the end of the day, teaching math is more like science, engineering, business (practical activities using preponderance of evidence) than it is like math itself.
Coverage of old material: I would take a middle road when using topics that students are "supposed to know" but in reality often don't remember. Don't just run remedial sessions and reteach the topic in general form. Don't just expect students to have ready facility.
Instead, something like (class before, end of class): "OK, for the next class...we are going to need to use partial fractions...I know that was a frustrating topic for many people in calc class and you may not have remembered it, so please take a look back at it tonight, because we will have to use it next class."
Then when you do whatever derivation you are doing and the partial fractions come in..."here are those darned partial fractions coming back. I know it has been a while, so let's walk through the algebra carefully here." [Then do show all the steps...and for the problem of interest, not a general partial fractions review.]