In the AL, the DH position allows older hitters to preserve their career for several more years. Why will the National League not adopt the DH given this benefit?
There are compelling reasons to introduce the DH to the National League and compelling reasons to remove the DH from the American League. Most likely the answer to your question comes down to money.
By introducing the DH to the National League, there would be 15 more starting positions across MLB. The prototypical DH is a power hitter. These types of players usually command expensive contracts. NL team owners don't want to introduce the DH so they can avoid paying those contracts. AL team owners likely have a reason to keep the DH out of NL as well. If the NL were to introduce the DH, there would be increased demand for those types of players, inflating DH salaries.
The question points out that the DH can extend the careers of players. Old players make more money than young players, because they can become free agents.
Keep in mind that the DH was instituted in the AL to increase offense as a way to bolster attendance, and consequently revenues. Although offense is at a low point in today's game, right now, MLB is flush with money from TV deals, so that pressure does not exist.