TL;DR: USPS is currently facing major financial issues due to an unsustainable model and struggles to work as it is. Additional load from mail-in ballots would definitely add to the problem.
First thing: how many voters can we expect?
According to PennState University:
Around 138 million Americans voted in the 2016 presidential election.
From Business Insider.
However, those 138 million Americans only make up 58.1% of our
voting-eligible population (those American citizens over 18). From
United States Elections Project.
In Pennsylvania, it is estimated that 61.3% of voting-eligible
population voted, around 5,965,000 people. From United States
While these numbers seem high, our voter percentage is not higher than
the 2008 (61.6%) or 2012 (58.6%) election turnout. From United States
It is difficult to ascertain how going to an (almost?) entirely mail-based system would change those numbers, but assuming no change compared to previous elections, it would be around 130-150m mails.
As noted in the OP, the USPS (United States Postal Services) deals with 181.9m mails per day on average, so it would definitely require spreading the load over several days, but it appears to be manageable at first glance.
However, USPS is currently facing major issues.
Trump mentions "New York, Carolyn Maloney". While he does not clearly mention what the exact problem he is referencing to, QNS reported in April about USPS have serious financial issues and asking for help from the stimulus bill:
“I want to commend the brave men and women of the Postal Service for
all they are doing in the midst of this pandemic,” Maloney said. “The
Postal Service is holding on for dear life, and unless Congress and
the White House provide meaningful relief in the next stimulus bill,
the Postal Service could cease to exist.”
More recently, USPS also released a statement indicating that they still can't sustain their current model:
We are currently unable to balance our costs with available funding
sources to fulfill both our universal service mission and other legal
obligations. Because of this, the Postal Service has experienced over
a decade of financial losses, with no end in sight, and we face an
impending liquidity crisis.
As well as indicating that they are facing massive penalties due to overtime in processing, transportation and mail delivery:
[...] The Postal Service Inspector General issued a report
entitled “U.S. Postal Service’s Processing Network Optimization and
Service Impacts.” In that report our Inspector General indicated that
the Postal Service spent $1.1 billion in mail processing overtime and
penalty overtime, $280 million in late and extra transportation, and
$2.9 billion in delivery overtime and penalty overtime costs in FY
2019. Yet, even after incurring these additional costs, the Postal Service has not seen material improvement in our service performance
While mails meant for election have a special tag meant to expedite their processing, many ballots still end up missing:
Between 2012 and 2018, 28.3 million mail-in ballots remain unaccounted
for, according to data from the federal Election Assistance
Commission. The missing ballots amount to nearly one in five of all
absentee ballots and ballots mailed to voters residing in states that
do elections exclusively by mail.
[...] Although there is no evidence that the millions of missing
ballots were used fraudulently, the Public Interest Legal Foundation,
which compiled the public data provided from the Election Assistance
Commission, says that the sheer volume of them raises serious doubts
about election security.
In light of the problems USPS is currently facing, additional load from a transition to mail-in ballots would severely strain a system that is in fact already struggling.