Requesting ad-hoc small plates/tastings

First, I doubt any locals would order that way, so I imagine it would be more common in places frequented by visitors. Beyond that, using Rome as an example, pasta dishes are prepared fresh. A Cacio e Pepe is tossed together in short time, as is a Carbonara, and the other several Roman pastas. I think the dishes you would get in requesting a mix would be ones prepared well ahead That's not all bad, lots of other things, but think you are leaving yourself open to disappointment, or at least not as much choice as you would like. The other issue is trying to explain what you want, language is not usually a barrier, but that may take some communication.

I think smaller places where things are cooked to order would be less receptive, a larger tourist centered place would be, Eh, sure.

Lots disagree, but I love going to a restaurant in Italy, we always order a bottle of water, wine (either carafe or bottle), an antipasti, maybe two. Then, my wife usually gets pasta, I get Pasta or a Secondi. We add on a Contorno, usually a side dish of vegetables, and dessert and coffee is usually a toss up (we opt for gelato while walking usually), but if we do not get dessert, I usually have a Digestif, a liquor from the area

This is not something which normally happens in restaurants in Italy

I would be interested in seeing that video reference. We have come close to what you are asking. We have been to dinner several times with a former student who now lives in London. We always go to the same small, family restaurant and the routine is always the same. Never see a menu or for that matter - a bill. Jim is greeted warmly by the head waiter. After we are seated he asks to see XXXX. Don't know if he is the chef or the owner or both. Has the white hat of the chef. They will have a brief discussion of what he is preparing that evening and they agree on two or three dishes and some sides. All served family style with appropriate wine. Never have been disappointed. After a long and pleasant evening, we just get up and walk out. Have asked the waiter for the bill but he shakes his head no and says, No ticket. It will be a hard act to follow if he ever comes to Denver.

I remember that video too, but wasn't it just two half servings? That seems more reasonable...
From devour website: "Primi are not usually shared, so if you want less, you can ask the waiter for a mezza porzione (half-portion)."

Some restaurants serve "assaggio di primi" (first course tasting) which is usually 3, max 4, pasta dishes in smaller portions, also to share.

Primi may not be usually shared (according to whom?), but if you want to share ask for two plates and nobody will bat an eye.

I think you can get what you want asking "un assaggio di..." followed by primi/starters/main/side dishes. "Un assaggio di" means "a bite of" or "a tasting of". If they usually do it, they'll do it for you, otherwise they will just say no. A "tasting of starters" is quite common in Piedmomt's most traditional restaurants, but you'll see it all over Italy.

Sometimes you'll see "tris di primi" written on menus, that means "a tasting of 3 pasta dishes", it's nothing strange.

I see no problem in cooking fresh food and serving a tasting of food this way, you toss a little more pasta in the pot and warm up a little more sauce. Of course you'll get what other customers have already ordered, nobody's going to cook 50 grams of carbonara just for you. Time is money in the kitchen.

Primi are not usually shared

What? Where? When you are having lunch in a place (mainly if there is something new or unusual with respect to your standard food) with a friend or wife/husband is pretty normal order two different dishes and share what has been ordered.
If you are in a little group you can even order a "bis" or a "tris" or different kinds of pasta (two or three different kinds). Usually is necessary to be at least two or three people (depends by the restaurant). When you order a "tris" is often served with some shared trays left in the middle of the table and everybody take the ones (and quantity) who prefer from each tray. Is called "Servizio all'italiana" (Italian service): so if is called in that way, why somebody declare that share food is not usual in Italy?
Even if this kind of service is more common for pasta is done in several places for main course.
Is not done in high level restaurants (star rated or similar), where you receive only what you have ordered. But in that restaurants usually portions are small, so are just "half portions".

I remember something happened one month ago. I was in a restaurant in Modena with a family of Americans. Since they have only one day in our region and want to try several different dishes, we ordered two different "tris" of pasta (regularly written on the menu). Everybody received a dish with three little portion of pasta and there where two different solutions, so everybody in the family sharing their dishes had the chance to try six different kind of pasta. Was a very standard and traditional restaurant in the countryside: nor the waiter nor everybody else in the restaurant was surprised of that kind of order (even if I suspect that the chef mumbled a little bit to prepare 6 different pasta for four people...).
We are more concerned when people order cappuccino with pasta, but this is another matter.
BTW: that is even a restaurant who still do the so called "al gueridon" service of boiled meat who was a speciality of Modena fifty years ago and now is disappearing.

I haven't tried doing that myself but should you be traveling to Rome, there is a restaurant that caught my eye in an RS Italy book offering small plates that I bookmarked to consider (in the event my ‘go to’ is closed for holiday), Osteria delle coppelle. From the RS book, “a slapdash, trendy place, serves traditional dishes to a local crowd and a fun selection of €3 cichetti (small plates) that lets you enjoy a variety of Roman dishes as tapas”
Address: piazza delle coppelle 54

@Frank-for future reference where is the small family restaurant? Would love to have chef recommended for dinner.