You offer a very sympatric view of tech support. When I was in a similar situation, I could give all the steps that I have taken to resolve a problem. I would have talked to the user to define the problem, I would communicate with the user and have them try different solutions; I would provide a work around to the problem.
Hours and days between communications does not imply that anyone is working on anything. In fact, such long periods of response would be a reason for a review, if not worse. If someone takes this long to reply, they do not know what they are doing, they are seeking help from others. The ‘others’ should be helping the customer to solve this problem; we should not be training tech support staff for their jobs.
You sound like management or a tech support supervisor explaining to the president or ceo way their services are adequate. Busy is a problem that they are bringing on themselves. If it takes a week to solve my problem, or even to address my problem, we need a tech support department. Top management needs to know about the lack of support problem to get some good staff on board.
I realize that that this company does not offer telephone support. This company does not allow any personal contact between themself and any customer for any subject or problem. If I were an investor, I would be very concerned about the level of communication in this company within departments and especially this their paying customers.