Hi Harold and Erin,
I’ve been where you are — my son, who is now 9, was 18 months when diagnosed. We tried for 10 months to make it work with shots only (access to either a pump or CGM was out of our reach due to insurance issues) and it was misery, although our problem was highs, not lows. When at least we were able to get better coverage for him, we switched over to the pump. So, the pros: We saw much better control of highs IMMEDIATELY, and his A1c went from over 8 down to the mid 7’s, which is a huge improvement. About a year later, we got approval for a CGM and the benefit of that wasn’t so much that it helped improve the ability to sleep without stress — although it did — it was that we were able to see the patterns of blood sugar fluctuations during the day. Once we had that data, we were able to do something with the pump that you simply cannot do with shots: program the basal to increase during the times when more is needed, and decrease during the times when less basal insulin is required. That capacity, in a nutshell, is what makes the pump so useful — the ability to control the level of basal insulin according to time of day. Pumps also give you the ability to stretch a bolus out over time rather than give it all at once, which with toddlers can be especially useful because it means if they get halfway through a meal and decide to throw the food on the floor, you can shut off the insulin that was intended to cover what’s now being eaten by the dog 
And, as you noted, you can program the pump to give insulin in incredibly small increments, which is not possible with shots — and it does make a difference. Example from my experience: When Eric had been on his Paradigm 522 for 4 years, the pump developed a small crack in the reservoir housing. Since it was out of warranty, we applied to the insurer for an upgrade, and got him a new Paradigm Revel that, among other things, had the ability to program insulin deliver in increments as small as 0.005 units. We justified the upgrade by suggesting that more pinpoint delivery of insulin would help with blood glucose control — and quite frankly, I thought we were simply telling them something that sounded good but probably wouldn’t actually make much difference.
BOY was I wrong. It has made a HUGE difference. Eric’s A1c, which on the old pump was consistently in the mid-7’s, is now consistently in the high 6’s and low 7’s — WITHOUT FREQUENT LOWS. The only time we really get lows is when he has a higher-than-normal activity level during the day, and we can compensate for that with lower basals at night.
So that’s the pros. The cons: managing with a pump and managing with needles are not the same, and it does take a certain amount of mental adjustment to realize that you now have to assess the basal insulin’s activity along with the bolus insulin’s activity, and the ratio of one to the other, when trying to figure out preventive strategies for lows or highs. Also, carrying a pump around can be somewhat fatiguing for a small child (I solved this problem by sewing pockets onto the back of my son’s shirts, with a little slot for the tubing, so that it both made it easier for him to deal with by distributing the weight onto his back rather than one side of his waist, and kept the pump itself safe from exploration by curious fingers — and kept the tubing from snagging on stuff too.)
You do have to change pump sites every 2 to 3 days, but we put a blob of lidocaine/prilocaine cream on the skin where we plan to insert the site an hour ahead of time, and that numbs it so he feels nothing. My son just recently started changing his sites himself, and the fact that we could numb the skin (and the skin blanches so you can actually SEE where the numbed skin is) was of key importance in helping him get past the heebie jeebies about sticking himself with a needle. Your doc can probably prescribe this cream, it’s not at all expensive. The flip side, of course, is that you don’t have to do shots anymore unless you have a pump site that pulls out or gets crimped — and that’s another pump con, that the sites sometimes don’t work as advertised and you wind up with high blood sugars and ketones. But that’s why we keep syringes as a backup.
Hope that helps!
Oh, and… as far as learning this stuff from the doctors? Most of them don’t live with it day to day. They understand how the body works (or doesn’t) and the physiology of diabetes, but they are specialists, and their specialty is how to apply medical knowledge to the disease process — so unless they have a child with diabetes or have lived with diabetes themselves, they have absolutely no way of knowing the day-to-day knowledge accumulated by patients. Eric and I have been regularly giving seminars at a nearby medical school that brings patients with chronic illnesses in to talk to the first-year students to help them learn just what kinds of daily stresses face patients that most MDs and DOs never hear about, as a way of trying to pass on that knowledge. Nurses who are CDEs usually have a better handle on these things, and if you haven’t been given regular appointments with a CDE, ask if this can be arranged — I’ve learned more from the CDEs at my son’s clinic (one of whom is a type 1 diabetic herself) than from anyone else.
I am a big 80’s guy…big flashy colorful button up shirts with pockets. hehe…you can buy them on eBay - it’s where I get all my super cool (embarrass the kids and wife) shirts!
] talents, I’ve been known to make some dreams come true with my sewing machine and some fabric).