Whether διαθήκης should be translated "covenant" or "testament" can probably be deferred to another discussion presuming we have no disagreement that what is in view is elsewhere referred to as "the new covenant". This is a covenant that God said that he would make with the houses of Israel and Judah:
Jer_31:31 Behold, the days come, saith the LORD, that I will make a
new covenant with the house of Israel, and with the house of Judah:
Heb_8:8 For finding fault with them, he saith, Behold, the days come,
saith the Lord, when I will make a new covenant with the house of
Israel and with the house of Judah:
Since this covenant is specifically only with Jews and not with all men then "the masses" is inappropriate. "The many", then refers to the Jews or some subset of the Jews. In my view this is the 144,000 aka God's holy "remnant" who "first fruits to God":
Rom 9:27 Esaias also crieth concerning Israel, Though the number of
the children of Israel be as the sand of the sea, a remnant shall be
Rev_7:4 And I heard the number of them which were sealed: and there
were sealed an hundred and forty and four thousand of all the tribes
of the children of Israel.
The author of the appropriately named midrash "To the Hebrews" explains the relationship between the death of Jesus and the redemption of the "remnant" of the Jews like this:
Heb 9:15 Therefore he is the mediator of a new covenant, so that
those who are called may receive the promised eternal inheritance,
since a death has occurred that redeems them from the transgressions
committed under the first covenant.
Here I repeat the verse but with explanatory information in brackets:
Heb 9:15 Therefore he [Jesus] is the mediator [administrator?] of a
new covenant [testament], so that those who are called [the
remnant/faithful Jews] may receive the promised eternal inheritance
[eschatological rest in the middle east], since a [ratifying] death
has occurred [referred to in Matthew 26:28] that redeems them [the
faithful remnant of Jews] from the transgressions committed under the
first covenant [the law of Moses].
We can clarify this further with some formatting:
Therefore he [Jesus]
is the mediator [administrator?]
of a new covenant [testament],
so that those who are called [the remnant/faithful Jews]
may receive the promised eternal inheritance [eschatological rest in
the middle east],
since a [ratifying] death has occurred [referred to in Matthew 26:28]
that redeems them [the faithful remnant of Jews]
from the transgressions committed under the first covenant [the law of
The death of Jesus has a universal scope as a propitiation:
1Jn_2:2 And he is the propitiation for our sins: and not for ours only, but also for the sins of the whole world.
But as the "death brought in" to ratify the new testament it only has application to "many" and these "many" are the faithful remnant of the Jews who's transgressions committed under the law of Moses are released thereby.
That the author of "To the Hebrews" understood διαθήκη as a "testament" or "will" (which is what the Greek word would normally signify) as opposed to a "covenant" is in evidence in the subsequent passage:
Heb 9:16 For where a will (διαθήκη) is involved, the death of the
one who made it must be established. Heb 9:17 For a will (διαθήκη)
takes effect only at death, since it is not in force as long as the
one who made it is alive. Heb 9:18 Therefore not even the first
covenant (διαθήκη) was inaugurated without blood. Heb 9:19 For when
every commandment of the law had been declared by Moses to all the
people, he took the blood of calves and goats, with water and scarlet
wool and hyssop, and sprinkled both the book itself and all the
people, Heb 9:20 saying, "This is the blood of the covenant that God
commanded for you." Heb 9:21 And in the same way he sprinkled with
the blood both the tent and all the vessels used in worship. Heb 9:22
Indeed, under the law almost everything is purified with blood, and
without the shedding of blood there is no forgiveness of sins. Heb
9:23 Thus it was necessary for the copies of the heavenly things to
be purified with these rites, but the heavenly things themselves with
better sacrifices than these.
So to answer the question, "the masses" does not fit the context of Matthew 26:28 and should be considered inaccurate in that context.
Might the word "masses" fit another context? Perhaps.