Were they suddenly in a position to dominate Lombardy and pressure powerful, more distant neighbours like Venice, Florence, and Savoy?

Milan did enjoy a post-pandemic boost to its power, and went on a conquering spree. Not quite as far as Venice and Savoy, but Florence and Genoa were certainly in trouble. In 1349, after the Black Death had passed and Giovanni Visconti had consolidated personal control of the city and surrounding Lombardy, the influencing and conquering began. Bologna, which had lost around half its population in the pandemic, was in no position to resist when Visconti demanded lordship of the city and capitulated to him in 1350. Florence attempted to form an alliance against Milan, knowing that they could only hold out against them with help, but Visconti knew of Florence's manoeuvring and invaded them in 1351 before a decent alliance could be formed. In 1352, Genoa also bowed before Visconti. In 1353 the relatively minor but notably troublesome city of Novara followed. Within five years of the Black Death's passing, Milan's ruling family had extended their control over most of north-west Italy and their hold over Lombardy was strong.

The difficulty is how strongly those events were tied to Milan's remarkable performance during the Black Death. It is true that Milan did far better than their neighbours, and a lot of that is owed to the Visconti being a bunch of paranoid tyrants. Their intelligence network was turned from spying on rivals to monitoring the spread of the plague, and they imposed strict public health measures to slow the plague down. In particular, they tightly controlled the bridges in their territory to prevent the movement of people and lock down the Visconti domains while the disease ravaged Genoa. You can read more about that here in an answer by u/AlviseFalier.

But that doesn't mean Milanese conquests in the years after the Black Death passed were down to the pandemic. When Florence tried to put together a coalition against Milan in 1350, the Visconti percieved that as a genuine and serious threat to their power. They weren't thinking "no worries, they're mostly dead and we're not", they were thinking "this could be a problem, send messengers to get allies of our own". Florence's diplomatic initiative did not fail because was too dead from the plague to be thinking about military campaigns, it failed because Milan made a better offer to Florence's prospective allies. Although the pandemic did give Milan an advantage, and Giovanni Visconti seems to have known that by how quickly he mounted several military campaigns in quick succession once it had passed, that advantage was far from decisive. Milan's success from 1348-1354 did owe something to the Black Death, but it leaned no less heavily on more traditional networks of alliances between the Italian cities or the powerful families that often ruled them in this period. There is also the factor of Giovanni Visconti himself. Everything I have talked about was his doing, after he was given sole rulership of Milan. And when he died in 1354, it fell apart within a few years until another strong Visconti could put it back together again in the 1390s. It was not enough for Milan to simply walk up to a city, point out that Milan was in much better shape than them, and they would submit (except perhaps Bologna). Milan still had to play the delicate game of Italian politics, they still had to think carefully about their targets, they had to avoid serious military defeats, and they still needed leadership that was capable of doing all of this. Any advantages from surviving the pandemic in decent shape were not decisive.

So yes, Milan was suddenly in a position to dominate Lombardy and pressure their neighbours into submission, I don't think there can be much doubt that Milan's fortunes during the Black Death gave them an advantage. I also think the Visconti were aware of this given how organised the Milanese military seems to have been at this time, and Giovanni seized the opportunity he saw in front of him. However, that advantage had serious limits. When Giovanni Visconti died in 1354, whatever advantage the Black Death had given them evaporated. That indicates to me that if the Black Death did shift the local balance of power in favour of Milan and the Visconti family, those advantages were temporary and depended more on a leader willing and able to exploit the moment than on the moment itself.