Short answer: Yes, there's another important reason that government bonds are often considered money, which is the role they play as a store of value.
There are three dimensions of money that form the long-standing textbook definition of money (see Mankiw, “Principles of Economics, 8th Edition.” p. 605):
- Use as a medium of exchange (i.e., its value for settlement of claims).
- Use as a store of value (the degree to which it is a "safe asset”).
- Use as a unit of account (the degree to which it is used to denominate claims).
In the modern framework different assets are considered to fulfill these dimensions to differing degrees, which is why different measures of money include or exclude different assets.
Thus, the state of the art is not to think of assets as being either "money" or "not money," but to think of different assets as being more or less "money-like," or having differing degrees of "moneyness."
Dimension three, for example, is generally fulfilled only by physical cash and claims on banks. Claims aren't settled for "20 units" of Treasury bonds, they're settled for a certain par value of those bonds, denominated in dollars. So clearly Treasury bonds are not at all money-like along this dimension.
Dimension one, the one you raise in your question, is fulfilled by Treasury bonds to a greater degree than one might expect, though still only to a very limited amount. While Treasuries are not used for settlement in the real economy, they are used for settlement somewhat frequently in the financial system, which is responsible for a disproportionate amount of financial transactions (though even in finance they are used far less often than settlement for cash). For example, Treasuries are commonly used in securities lending transactions to obtain other securities, and they're often used to meet default and liquidity fund obligations to central counterparties.
Where Treasury bonds shine, though, is on dimension two, their use as a "store of value,” or as a “safe asset.” Per Gorton:
Safe assets play a critical role in an(y) economy. A “safe asset” is an asset that is (almost always) valued at face value without expensive and prolonged analysis. That is, by design there is no benefit to producing (private) information about its value. And this is common knowledge. Consequently, agents need not fear adverse selection when buying or selling safe assets. Safe assets can easily be used to exchange for goods or services or to exchange for another asset. These short-term safe assets are money or money-like.
The “short-term” part of this refers to the fact that their usefulness as a store of value rises as their residual maturity (and thus their duration, or sensitivity to changes in interest rates) falls. Once their residual maturity falls below three months, they're treated for accounting purposes as being equivalent to cash, because their value isn't likely to change significantly in response to changes in the interest rate environment (this is true under both US GAAP and IFRS):
Cash and cash equivalents comprise cash on hand and demand deposits, together with short-term, highly liquid investments that are readily convertible to a known amount of cash, and that are subject to an insignificant risk of changes in value. Guidance notes indicate that an investment normally meets the definition of a cash equivalent when it has a maturity of three months or less from the date of acquisition.
Importantly, Treasury bonds solve a very real problem for many firms that have large cash balances, which is that for large-value claims, they are actually more money-like than bank deposits. This is because there's a limit on deposit insurance (currently in the U.S., \$250k per bank, per account type, of which there are two, so \$500k per bank). As a result, for firms that hold large amounts of cash, bank deposits become a relatively poor form of money, because they're not eligible for the government guarantee. As a result, if the bank fails, they're exposed to both credit and liquidity risk (they may not get all their money back, and even if they do, it may not be available for quite a long time).
This is actually why firms frequently either hold balances in institutional money funds that do repos against government securities, or lend them in repo themselves— it allows them to upgrade what would otherwise unsecured claims on banks to claims that are secured by Treasury bonds, highlighting the "safe asset" role that these bonds play.
A very good summary of this concept, the "hierarchy of money," can be found in Pozsar's "Shadow Banking: The Money View." As he discusses in the paper, for many financial market participants, safety (dimension two) is more important than settlement value (dimension one) (emphasis mine):
[T]here is a strong case for the introduction of a new set
of monetary aggregates that track the supply of money and money-like claims held not for real
economy but for financial economy transactions. The Federal Reserve’s M2 aggregate measures the
money demand of households and has been used to analyze growth dynamics and threats to price
stability. Because the bulk of money claims included in M2 are insured, it was built following a
hierarchy based on transactional liquidity. But for institutional cash pools, money begins where M2
ends, and as the crisis has shown, intra-system holdings of uninsured money market instruments can
pose threats to financial stability. Institutional cash pools hold money claims mostly for portfolio
management reasons and because they are too large to be eligible for deposit insurance, their focus is
on money claims’ safety – that is, proximity to the government – first, and transactional liquidity
This is, of course, why the Fed had to create the Reverse Repurchase Program. Without it, cash lenders in repo were occasionally so eager to exchange their "cash" in the form of claims on banks for claims on Treasuries that they were willing to accept negative real (and, in certain circumstances, negative nominal) yields on general collateral repos against Treasuries— they were literally willing to pay to hold Treasuries rather than bank deposits. As a result, the Fed set up the RRP to allow their counterparties to do unlimited amounts of repo at a fixed rate, setting a floor under short rates in money markets.
Finally, there is a good working paper out of the Federal Reserve, authored in part by former Fed governor Jeremy Stein, that goes into further detail on the use of Treasuries as money-like claims.