I ride 3 miles to daycare, followed by another 9 to the office. Moderately hilly in the Denver Colorado burbs. Prior to that, I worked from home with a 7 mile RT (twice a day) slog to daycare in flat New Jersey, similar climate.
TLDR version: Get a midtail/longtail bike, or a single rider (short tail?) bike towing a Weehoo. Ditching a Burley/Chariot at daycare is a pain.
In Jersey, I loved the Burley Cub trailer. I rode an old hybrid bike. The Burley stayed attach for the daycare round trips. If I went to a store, I generally brought it in with me as a stroller/shopping cart. This worked for humid summer heat, rain, sleet and light snow. The commute included a half mile of bouncing through a ditch and farm field. The Burley is tops for weather protection. My daughter and I far prefer a longtail though, because communication to a trailer is pretty much impossible without stopping. She still prefers the Burley if it is very wet or cold, though.
I tried the same here, but locked up the Burley at daycare. This is a total pain in the butt. It takes 3-5 minutes to unhitch, remove burley wheels, and lock everything to a fence. If your daycare allows you to store the Burley inside a secure area, might not be so bad. Mine fairly said that they had no room, but I could lock to the outside perimeter fence along the sidewalk.
A rear rack seat works really well for a 3 year old. It still works well for my large 4 year old. On a normal bike, this does mean that you have no rear rack pannier space though. So, you would need a backpack or a front basket. My front basket attempt failed due to instability above 15mph, in retrospect I had some install issues.
My family has 2 dedicated kid carrying bike solutions. They are much faster total trip time, because I do not have to mess with the Burley. An option that I did not explore enough was using a regular bike towing a Burley Piccolo/Kazoo, or Weehoo trailer bike - those would lock up much faster than a Burley, because wheels could stay on. I would recommend the Weehoo for a 3 yr old, towed by whatever single rider bike your prefer.
One is a Bike Friday Family Tandem, with the rear stoker seat setup for my kiddo. This was viable starting at about age 3.5 for her, with some modification to rear seat, bars, and crank. I later electrified it to cope with the hills, but on flats I could easily hold 16mph unassisted with this rig, and it only weighed about 45lb before I messed with the electrics. Bike Friday also makes a Haul-A-Day cargo bike, which can be electrified. Its a cool option - lightweight, compact, low step frame. The tandem gets a lot of appreciation from folks who see my kiddo and I riding to school.
The second bike is an Edgerunner Swoop. This also started off pure mechanical, but got an electric kit. This bike was a little slower than the tandem, both unassisted. Due to weight, but also the air drag of the rear "kid cage" setup. Now that both the tandem and the Edgerunner are electric, I prefer the Edgerunner. However, I am able to park inside at work. We hesitate to leave it outside in a rack all workday, and we live in a decent area. As a kid and grocery hauling commuter, this thing is awesome. HIghly recommend an Xtracycle Edgerunner eSwoop. Get the Swoop frame, the step through is really nice to have. An eSwoop kitted out for kiddos would run about 5k.
IF you are below 5'10", the Yuba E-boda boda step through looks awesome. The newest model with the tire hugging rear rack. The stepover frame is less attractive - I am fairly young and limber and it sucks to throw a leg over the top tube with a kid in back. My kiddo got a foot to face several times when riding jumpseat on my hybrid.
The Tern GSD was not out yet when we went shopping. This bike is truly awesome. The latest release has nuvinci as an option, although it retains chaindrive and a derailleur style tensioner. If you are ALOT lighter than me, consider the Tern Vektron - but the 240 pound total payload including rider is a deal killer for me. For someone at or below 160 pounds, this would be a hella versatile rig.
Bucket bikes are really cool. But a regular longtail can be hard to park, a front bucket bike would be hell. You dont really need the weather cover in Palo Alto, and the cover options are one of the huge pluses of bucket bikes. A longtail is also a much easier transition from regular bike.
Consider a lower spec bike - theft is a problem that deters from using the bike for errands. It is multiplied for high value bikes. 7 miles in flatland is actually not bad for a standard, unassisted bike; which is alot less likely to get nabbed. I am jealous of your bike lanes, I was totally envious this summer when I saw the green painted lanes throughout Palo Alto.
Check your local laws regarding Class 1/Class II 20mph limit bikes vs Class III 28mph bikes. Around here, going to class III means no bike paths for you. Bike paths can be a huge time saver or stress reducer vs the street.
