I'm a bit curious about the 21 hotels in 48 days. Do you mean 48 nights? In either case, that's a lot of moving around. If it is nights, that's only about 1 day in each place and it seems like your trip will be more about being on the road or rails or plane than about being in the places you are going. All that extra transport will cost more, too, whether you rent a car, take the train or fly. You have lots of time off the boat and on the ground on this trip. You might enjoy it more if you slow down a bit.
Having said that, I make all our hotel, B&B and apartment reservations using a variety of sources to find the places. Then I book directly with them rather than going through any of those sources. If anything goes wrong, I prefer dealing with the place directly over going through any go-between. So far, nothing has ever gone wrong.
I would definitely think your wife's research on this is correct. Your budget is pretty high for daily lodging costs (182 euro today), so if you care about saving money, that's another place you could shave even more off your costs. If you decide to slow down, apartments are a great way to enjoy "living" in a location for a few days and they are often cheaper while providing more freedom.
The main sources I use to identify lodging are Booking.com, Rick Steves' books, and Trip Advisor. The sources I use to identify apartments are HomeAway and VRBO (Vacation Rental by Owner). Those two are affiliated. I have used Airbnb to identify apartments to rent, but have not yet rented anything that way. There are many other options for finding apartments or hotels in specific areas or cities. For example, and especially for small towns and if you are driving, we have used Logis in France (http://www.logishotels.com/en.html).
It would not be that difficult to book your own trip, but doing this is work, there's no doubt about it. Because lodging has consistently been our highest expense (not counting airfare) over the last few trips, I try to find the best place I can for a reasonable price. We average about $150 per night for lodging (109 euro today), but that's an average. Some places are cheaper and others more expensive. So far we have never stayed in any bad places, or in any American-style hotels. And our apartment in Amsterdam (above Tiffany's) last April was amazingly beautiful at a very reasonable price.
One thing that we gain by booking on our own is that we actually get to see the rooms or apartments I am booking, so we can see what we are getting. My husband in particular has some requirements that are deal-breakers if they are not present: a shower with a shower head high on the wall (as opposed to a tub with a sprayer and no place to hang the sprayer up high) and an available plug close to the head of the bed for his CPAP machine. You cannot search for these things and I doubt that any travel agent would bother checking the lodging to see if they are present. The only way I've been able to determine if a room or apartment has them is to look at the pictures.