Directly apply to a junior high school

It is possible for an ALT to directly apply to schools in Japan and effectively cut out the middle-man?

Generally speaking about public schools, no it is not possible. While there are plenty of direct hires working in Junior High Schools in Japan, they typically are special arrangements (sister city exchange for example) or the result of a previous relationship between the two parties, for example I know of a few alts that work directly for their school but only because they had been working there as a dispatch or JET previously. Of course there are exceptions, but this seems to be the predominant situation as I am aware of it.

To work directly for a school as a English teacher in Japan you need to have jumped through various hurdles that ALT's do not normally need to worry about.

When you join as a ALT through a program like JET this is all taken care of you and you do not need to worry about normal teaching qualifications normally required. Same thing applies when applying through the local city, they take care of everything.

Without these special procedures in place for English teachers you will be required to produce things like: Statement of professional standing & your teaching certificate that the local prefecture awards you. Basically it's a background check that the school itself is not capable of doing.

No sane person would go through those procedures just to teach English when they don't need to as you will be processed as a normal teacher.

Like yllwsmrf says, exception being if this procedure has already occurred in the past (ex JET)

Just to add a little more, ALTs who assist and are supplementary to regular licensed Japanese teachers do not require teaching licenses to be hired. They would just need to qualify for the regular work visa.

Alternatively, some boards of education employ special foreign instructors who are allowed to teach classes by themselves and generally have all the responsibilities of Japanese teachers, but are given a special exemption and not required to have a teaching license. These jobs also typically hire from within and would not be available to a newcomer to Japan without an introduction or prior relationship with the hiring organization.