I think you are asking two questions: 1) how does the engineer justify adequacy of the investigation program, 2) how does the engineer justify costs. The answers would get long and complicated if I had enough time.
Public works projects are usually assigned to the engineer through a basic ordering agreement (BOA) or similar, or the engineer could be part of a design-build team. In each case, the engineer is required to prove that they are qualified by a process of quality-based selection (QBS) process. So, there is a big element of trust but only after the qualifications have been proven. The qualifed consultant would develop the investigation program after qualifications have been established.
We don't talk about costs until after each consultant has proven their capabilities. It is rare for a public client to take issue with the cost since this could comprise the integrity of the program. The more common problem is not proposing enough investigation (i.e., the engineer is too shy to ask for what the job really takes).
The quantity of borings and samples is usually justifed by established guidelines. For example, AWWA has boring density guidelines for water tanks or state agencies have established boring density for bridges. Sample density is also often prescribed. Additional sampling is often required is specific structures are being constructed such as a tunnel or a certain geologic formation is encountered (bedrock, soft clay, peat, etc.). Other investigations may be warranted such as geophysical if the engineer determines that its worth it and makes his case to the owner.
Most public agencies have reviewers whose only purpose is to review the adequacy of geotechnical work. I do sometimes see statistical analysis used to evaluate test data but never to review the adequacy of a proposed test program. When statisical analysis was used to evaluate test data, it was only for the largest projects where there was a lot of controls in-place for QA/QC of the geotechnical work. I have done confidence intervals on smaller projects but only to amuse the client after they requested it. It had no impact on the outcome for the foundations and earthworks.