Delta is a very poor approximation of the risk neutral probability that the option will expire in the money.
N(d2) is the actual probability, which eventually becomes zero for very long dated options and / or when there is very high implied vol. On the other hand, delta (N(d1) if undiscounted forward delta), becomes 1 (100%) in this case.
You can find some details here.
Edit
N(d2) simply is the measure for the risk neutral probability of exercise. It becomes zero because of the way the model is built. The higher σ (Vol / sigma), the more the global maximum of the probability density function (the mode) shifts towards the lower bound of the lognormal distribution.
The reason the option value of the option itself is approaches the discounted value of the forward is that more and more contribution of the expected value come from values of ST when ST≥K as σ (sigma) grows, although the probability for those values to occur goes to zero. So, we have a sort of competition of limits, where the values of ST above K increase faster than their probability to occur goes to zero. It is mainly a theoretical result based on basic probability theory because very high vol and very long tenors are unlikely in option pricing.
Nonetheless, Warren Buffett explained his take on the Black Scholes formula for long-dated options (which has a similar effect as large IVOL) in his 2008 letter to the Shareholders of Berkshire Hathaway.
The Black-Scholes formula has approached the status of holy writ in
finance, and we use it when valuing our equity put options for
financial statement purposes. ... If the formula is applied to
extended time periods, however, it can produce absurd results. In
fairness, Black and Scholes almost certainly understood this point
well.
Below is a gif I made in Julia showing this impact (you need to compare the mode vs the mean).

Edit 2
That vol is not constant and returns are not normally distributed has very little to do with this question. If you get the math, you get the result. Delta is the probability of the option being ITM under the stock measure (this is yet another equivalent martingale measure which uses the stock as numeraire). N(d2) is the probability of the event {ST≥K} in the risk-neutral world. That is always the case.
Insofar, for the question at hand, N(d2) is the only correct answer, and N(d1) only approximates it (which the answer referencing Natenberg points out correctly) since time to maturity and volatility are typically small numbers, i.e., d1=d2+ σ*sqrt(T) ≈ d2.
Regarding the concern that vol is not constant, and returns are not normal; option markets correct for this with the IVOL skew - which affects N(d2). See for example this answer for an interactive chart showing how implied vols correct for skewness and kurtosis in the return distribution and how to fit vol smiles with SVI (one way to do it).