"Unfortunately, it's hard to force men to take their leave"
There's no reason to force anyone to do this.
The statement is dystopian. If you offer it to people, and they legit don't want to take it, anything beyond that you're the problem, not the solution.
I have worked in law and tech. I have seen many men who took hardly any time off in law firms because it was heavily frowned upon. In tech, it is much less frowned upon, as evidenced by the fact that many big tech companies offer the same leave to fathers as mothers.
I believe the reason for the different behavior is not that lawyer dads don't care about their kids/wives as much as techie dads. Rather, I believe it is because in the cultural context, they feel pressure not to take much leave. So by putting pressure on companies to make the option a real one for fathers, I think we could make fathers happier (and kids/mothers also) by allowing them to choose the option that they would want if they could do so without repercussions.
Also, there is something to be said for leveling the playing field for mothers. So while I will agree that we don't want to force people to do anything (and my suggestion actually does not force anyone to do anything), there can still be valid reasons that a society would want to incentivize people in certain ways for the good of the society as a whole.
In The Netherlands a similar law was put in effect in July 2020. I'll reap the benefits of this very soon (my partner is due tomorrow). I don't believe its fully paid though (IIRC 70%). Before this it was two days. TWO days!
This is our second child, and from the first one I know I still had lack of sleep the first 48 hours. The first week is the roughest (especially for two autistic parents an absolute hell with lack of sleep plus all kind of new stimuli), with the following months almost as severe. I believe it is incredibly woman unfriendly (I'd even call it misogynist) to give men no time from work.
On the father and mother this leads to less fatigue, burn-out, and on long-term better output. For the child, I believe its difficult to see the effect on this on short-term. For example, the effect of father-child bonding might not be seen on the short-term, and its also no guarantee, so you'd end up with some kind of correlation in a study. Proving a causation on this is going to be difficult.
I think the issue isn't that someone may be forced to take the leave when they don't, rather there are hidden incentives for them to not do so when they do. There's a heck of a lot of room between the latter and the former.
On another note we are seeing fertility levels fall in many countries to the point where it's becoming an economic hazard. Parental leave thus becomes a social good. A smart market friendly solution, that would also be a win for families, would be to do more in terms of culture to encourage fathers to do this. In Japan, I've noticed high ranking male ministers almost uniformly taking parental leave to set an example to emulate. I'll add that such an initiative should still leave freedom for people to make their own choices for their own circumstances.
Nudging parental/father's leave in order to increase fertility levels seems like a drop in the bucket... Why not tackle the real issues? Pay people more, tax them less (lower income taxes, increase capital taxes), make property cheaper. Yeah that includes a good parental leave pay (possibly more than 100%, in order to specifically encourage fertility in highly educated women/couples). That will have 100x greater effect in terms of fertility, in addition to being a "social good" in other regards.
All those things you listed involve difficult trade-offs.
Increased capital taxes would on the margin decrease capital expenditures, and thereby hurt income growth in long-term due to lack of productivity gains. That's the last thing we'd want if our goal was to increase income. IMHO both income and capital gains taxes should be cut substantially.
Cheaper property necessitates more property, and at least here in US not building in the suburbs has become presidential campaign fodder. The most economically productive cities like SF have some of the most complex zoning and planning regulations that make boosting supply challenging.
If I had to guess what the right answer would be, it'd be higher minimum wages + stronger unemployment benefits that are designed to get folks back to work. Germany's post-Hartz system is an example of the latter. Higher minimum wages would force people out of work, but they would also motivate higher cap-ex in automation. Higher labor costs would make certain overly expensive automation projects more tractable to corporate management. The unemployed would get an opportunity to retrain for better, higher paying jobs and do so while having basic needs met.
You can probably see though how this solution is basically impossible in the current political climate.
There's no reason to force anyone to do this.
The statement is dystopian. If you offer it to people, and they legit don't want to take it, anything beyond that you're the problem, not the solution.