Tutoring is depressing because my students are struggling too much with my exercises. What do I do?
What you're experiencing is pretty much what all college teachers are wrestling with all the time everywhere. 

Allow me to point out that the abyss of need can go much, much deeper than what you're currently dealing with. In the U.S., most college students are attending a community college. And most students entering U.S. community colleges do not have 8th grade algebra skills, nor even 6th grade arithmetic skills (e.g., fractions, proportions, negatives, estimations, times tables). I've had a good chunk of a college math lecturer career at this point and I've never taught anything as high level as matrices. 

Here's how I've been able to jiu-jitsu it in my own mind. Think of it like a diagnostician. Doctors have to find it fascinating to track down hideous diseases. Detectives have to find it compelling to decipher ostensibly horrible crime cases. So too: I find it endlessly fascinating to diagnose what exactly is the block or problem in students' heads, or to track down the place where their reasoning first went off the rails. Ask questions to try and find where their reasoning starts and where it stops. Do something of a binary search to try to narrow down the problem spot. I'm continually discovering new gaps in people's knowledge that I never would have expected. It's surprising and amazing every day. 

Usually I do find that students simply cannot remember starting definitions. If they've come up through a system that depends on raw faith and memory (and not mathematical reasoning), they don't get how important that axiomatic basis is. Many also clearly have learning disabilities that make this difficult for them. "All your answers are back in the definitions", I say several dozen times per semester. Have the book handy so you can turn back to the starting definitions at any time, and see if they recall or can see how their reasoning got off track that way. In fact -- this explains why at your current school they're asking students to reiterate definitions; whereas for us that's natural and obvious, for some it's an overwhelmingly difficult blind spot to fill in.