2_i 

Interesting. Let's do the numbers: 5.5*1609m/3600s*0.11*110kg(?)*9.8m/s^2=290W just to lift yourself and the bike up at this speed against gravity and not counting regular friction losses like in riding on a flat. An elite racer at the peak of career is capable of 400W mechanical output on a continuous basis and normal fit people are at 100-150W. Somehow the numbers do not stack up.

cpach

I'd do the chainrings first if at all possible, as really most recreational riders who have a reasonably fast cadence don't need a 52t chainring very regularly (yes, I know you can pedal it down a hill, but you'd probably be faster focusing on tucking at that speed blah blah blah). If you need only a small change in gearing, switching to 50/34 rings would be fine, but if you're going to the trouble consider getting a subcompact crank like a 48/32 or Shimano's GRX 48/31 crankset to get really meaningfully lower gearing, with a top end that is honestly still entirely fine unless you often sprint in group rides. Don't just switch the small ring, you'll get pretty bad shift performance as the rings are intended to work in sets.

The cassette is probably the cheaper option, although you're likely to need a new chain. Max spec on that derailleur is 34t, but you're very likely to be able to shift a 11-36 OK, although you may risk slightly slower shift speeds.

DaveSSS 

Where did 110kg come from? My bike and I weigh 70kg. I'm 67 years old and can't put out as much power as I could 15 years ago, but I did ride the 28 miles from Idaho Springs to the top of 14,000 foot Mt. Evans with a best average speed of 11 mph. That’s slow compared to the record that's about 16 mph, but I'm no pro and I was 53 years old. Back then I could put out over 200 watts. I ride 10% grades on nearly every ride and I've measured the grades with a digital level. Substitute 70kg for your 110kg and you get 184 watts.

2_i 

I put 110kg hypothetically. Even with 184W, this does not stack up, as it represents 100% efficiency. Of course momentary surges of power are normal but this does not represent riding at certain speeds for certain grades. Fantasizing is human while superhuman physical effort is not. However, fantasizing does not add to discussion.

DaveSSS 

There is certainly nothing super human about my riding up a 10% grade at 6mph. I get passed by young guys frequently. I don't understand what you're implying. My computer speed is accurate and my cadence is between 65 and 80 on tough climbs. The Mt. Evans route only averages about 4%, hence the much higher average speed.

gsa103

184W (2.6 W/kg) is a perfectly reasonable effort for a fit life-long cyclist, especially for something like a 10-20 min effort up a known climb. That may be an above threshold value, but if they can recover on a 5% section, it's perfectly reasonable. That would be an FTP of around ~2 W/kg, not exactly Lance territory. Most people will go over threshold for steep sections of climbs, planning on recovery later. That's the advantage of knowing the terrain. On my regular climbs, I know exactly how deep I can go without burying myself.

2_i 

184W is without any regular losses in riding. You can add 50W for the rest - could be more when the drivetrain is a the limit of operation.

gregf83 

5.5mph up an 11% climb is about 970VAM which is manageable by most of my riding friends and we're all over 55. The steeper the hill the higher the VAM possible, provided you have the right gearing. I don't climb many 11% hills but here's one from the Malibu area a couple of years ago. Not a particularly fast time (132/498) for 55-64 yr olds on that segment.