Italy would be my choice for this, as it seems the Italians still dote on any children, especially the tiny ones. They'll get respect / attention they're not used to in restaurants, plus they'll have the excitement of gelaterie and pasticcerie which taste as good as they look. Pasta and pizza are tactile / visually stimulating for children, whilst you can explore the wider range of Italian food. I swear one reason Italian kids are often well-behaved is because of all the attention they get - no need to throw a tantrum if people are already paying you attention.
My sister and B-I-L took their 1 year old to Ravello and used one of those chest harnesses, and it was a very good move. The 1 year old even attracted the attention of a young Italian boy who came over with a daisy for her - they train them young over there!!! They had a great time.
So yes, but the destination does warrant some thought, as does the accommodation. However the time of year you are planning is perfect, probably our preferred time to go, allowing for town/city or rural or coastal or hill/mountains. Rarely baking hot, and rarely cool/cold. Often 'just right' as Goldilocks would say.
For accommodation, I'd make a strong steer towards an apartment to give you great flexibility, plus a little more space if tensions start to fray. It allows you to feed the little folk when they want, from the 1st meal of the day to the last. It will require more interaction in normal Italian shops / supermarkets / markets to pick up stuff, but take the kids and they'll generally break the ice for you. Also most apartments have a washing machine (do check) which helps reduce the amount of clothes to pack - my sister commented how little she could pack for herself, once all the kids stuff was packed - and that was only one child! Getting a place with a pool might excite the young ones, allowing one of you to potter around doing stuff, whilst the other keeps them splashing around safely.
Now for location, I'd definitely avoid Rome, Venice, Florence, as they are very busy with foreign tourists and won't be as obviously accommodating to the children. The crowds may be worryingly bustling. I'd probably avoid all the major cities as well, so no Bologna, Torino, etc. If the children get excited about the sea, then plenty of options but Italian coastal 'resorts' are a little different (not for me, but they seem popular), but there are places that have great appeal that are on the coast.
Cinque terre is an option, but some of the trails will be too much for a 5 year old, however the ferry or train will get you around very easily. It's visually stimulating and will seem a little 'fairy tale' in it's style.
Personally I prefer the Amalfi coast o CT, and for convenience I'd suggest Amalfi, though Sorrento is worth considering. Ravello is heavenly, but would require the mindset that you'll relax and take it easy up there most of the time, taking in the stunning views, villa cimbrone gardens and having the kids play in the square in the evening. Amalfi/Sorrento make it easier to venture to other places.
Rural Tuscany / Umbria has plenty of appeal and there are large numbers of villas to hire in the former, plus the advantage of English being widely spoken (as indeed it is in the above two locations)
Puglia offers something a little different and places like Martina Franca, Locorotondo, Alberobello etc. have unique style. Inbetween the towns the driving is easy, though it can be a little stressful in them, as the roads are sometimes narrow / twisty with quite a few now one-way. Trani near Bari airport has a good castle / coast, but for a stunning castle, head up to Castel del Monte.
There are many, many more options!
So not mad at all, though take advice on the kiddie logistics, and also try to keep the location logistics simple - ideally the whole holiday in one base location to avoid a stressful transfer.
regards
Ian
p.s. If there are any fevers, colds etc. whilst the kids are there, just about every farmacia in Italy has someone who speaks good English, and there are many of them, often with the usual rota of late opening. Everything else is stocked you'd expect in shops/supermarkets etc. but the erboristerie are a useful resource as well with a strong history of herbal remedies.