You can request whatever you want: I can certainly ask Home Depot if they will let me take a new fridge home for free - the answer rightly would be "No." As would the answer, here.
At the time of cancellation, your flight was on time as scheduled, and for whatever your reason, you had to cancel, so did so. At re-booking, you'll need to pay the re-issue fee. The delay later was not a factor in your decision. To me, asking for (and receiving) a refund would be fraudulent.
Let me preface this by saying, a few years ago now, I was booked on a flight, and about two days prior, I found out I had to cancel. This was an XXX-ORD-YYY connection. My wife was going XXX-ORD-ZZZ, on the same first flight. She wasn't canceling. I decided to hold out on my cancel - for one, I figured it would be a good chance to use the BP to go to the gate and then say goodbye there - on another, I saw the XXX-ORD flight was full and/or overbooked, so wanted to see if I could get VDB'd and refund anyway. Turned out they didn't need volunteers - waited right up until almost departure time, and called and canceled. About an hour later, the ORD-YYY flight I had been booked on started getting delayed. An hour or two later, it outright canceled. I was annoyed - obviously, would have been really great if I could get a refund for a trip I needed to cancel anyway (or they needed a VDB). But the point is I knew there was no justification for them to refund it fully - I needed to cancel - then did, irrespective of UAs cancellation. Their cancellation happened afterwards.
Sometimes, things in life suck, and that did, as does this for you (in a, might I add, very, very first-world problem way). Sometimes, you win, and sometimes you lose - in this case, you lost (and only sort of - because on the day earlier or later - it probably would have just been part of life). So my advice is to take it like that, and move on.
On another note, some in this forum would say this is one reason why you should wait until the absolute last moment possible to cancel a ticket - as things like this do happen. If your cancellation was done after the delay - you would have been entitled to the refund (as noted by another poster above).