Looks like on the server side this can be mitigated somewhat by the MaxStartups¹ setting for OpenSSH or equivalent behavior for other services that support SSH auth (e.g., Git forges like GitHub):
MaxStartups
Specifies the maximum number of concurrent unauthenticated
connections to the SSH daemon. Additional connections
will be dropped until authentication succeeds or the
LoginGraceTime expires for a connection. The default is
10:30:100.
Alternatively, random early drop can be enabled by
specifying the three colon separated values
start:rate:full (e.g. "10:30:60"). sshd(8) will refuse
connection attempts with a probability of rate/100 (30%)
if there are currently start (10) unauthenticated
connections. The probability increases linearly and all
connection attempts are refused if the number of
unauthenticated connections reaches full (60).
So it looks like it's possible to support ControlMaster while still somewhat hampering mass-cloning thousands of repos via SSH key without reauthenticating.
Admittedly I'd put this more in the category of making endpoint compromise easier to detect than that of actually preventing any particular theft of data or manipulation of systems. But it might still be worth doing! If it means only a few dozen or only a hundred repos get compromised before detection instead of a few thousand, that's a good thing.
Besides all that (or MaxSessions, as another user mentions), if an attacker compromises a developer laptop and can only open those connections as long as the developer is online, that's one thing. But a plaintext key that they can grab and reuse from their own box is obviously an even sweeter prize!
"The SSH key on my YubiKey is useless to attackers" is obviously the wrong way to think about this, but using a smartcard for SSH keys is still a way to avoid storing plaintext secrets. It's good hygiene.
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https://www.man7.org/linux/man-pages/man5/sshd_config.5.html