The cynical, pessimistic reason is that wildcat strikes have no real constituency. Obviously employers are against them; they're against strikes in general. Further, they aren't illegal in the sense that people go to jail for them; they're illegal in that people don't receive any special protections if they strike. I.e. employers can choose to fire workers who wildcat strike. Since they can also choose not to do so, this is unabashedly something of which employers would approve.
Less obviously, union leadership doesn't particularly like wildcat strikes. The reason being that such strikes bypass the union leadership. If people could get the same protections without the union leadership, they might choose to that. After all, union leadership costs money. A side issue is that union leaders have more credibility as negotiators if they can promise their workers won't strike if their demands are met.
The actual union workers who might wildcat strike won't care unless they actively expect their union leadership to fail them. Most of the time, this particular option isn't used and thus doesn't matter. As a general rule, the people who actively understood it would have been either the government, the union leadership, or the employer.
More optimistically, it adds stability to the system. Because all strikes have to go through the union leadership process, discussions between union leadership and employers can establish a certain amount of negotiating respect. If employers thought that employees would strike even if they gave the union leadership what they wanted, then employers would be less likely to make their best offer. They'd lowball to give themselves room to offer concessions.
The National Labor Relations Act sets out how workers are protected when they consider unionization. Wildcat strikes aren't covered by those protections, as they don't follow the process set out for protected negotiations. So employers are free to treat people who engage in them as if they simply chose not to show up for work.