Is there no room to issue a directive with halakhic force (on the part of the rabbinical courts, for example) so that, as part of halakhic intervention in family life, fathers too would be required to take paternity leave? After all, if the commandment of “be fruitful and multiply” is an obligation placed on men, isn’t there some point in giving it concrete content in private life?

Directives of the rabbinical courts have no halakhic force whatsoever. These are institutions appointed by the state, and they may perhaps have legal force (to the extent that the law grants them authority in this area). Nothing more.
Giving concrete content to the commandment of “be fruitful and multiply” is of course appropriate, and each couple should decide for itself how to do that. Why should the state or Jewish law get into the couple’s innards and obligate them in all kinds of things? Today there is an option for either one of them to take leave, so they should decide on their own.

If the state represents the public, then don’t they have authority by virtue of “they accepted it upon themselves”?

That’s not exactly “they accepted it upon themselves” (which is said regarding witnesses and judges). It is the law of the kingdom, or the authority of the town leaders. In that terminology, there is also halakhic force to a law that forbids crossing on a red light or requires paying taxes. If that’s what is being discussed, then it’s trivial. But the discussion was not about that. The state is not supposed to, and does not purport to, deal with the observance of commandments (such as “be fruitful and multiply”).