Agreed and some very elite companies have addressed this by casting a very wide net through the use of technology.
For example, per their website, anyone can apply to take the Jane Street placement exam as they don’t have strict requirements regarding school background or prior finance experience; however, the exam is designed for highly skilled individuals with strong mathematical and computer science knowledge, making it extremely challenging for most people without a strong foundation in those areas to pass. In other words take your shot and let’s see what you got.
They want to get the best and brightest and are well known to recruit heavily at MIT among several target schools but will allow any and all into the process if they can prove themselves in advance.
It is however a very challenging prescreen that must be cleared before you are actually even considered by an HR professional for an interview process that typically has 4 rounds of escalating difficulty and typically includes 15-20 current employees.
Versions of this prescreening (as @blossom suggests) are undertaken by most large investment banks who use technology to sort through the tens of thousands of resumes received. Key words, target schools, threshold stats, etc are all considered and used to cull down the proverbial heard. As OP seems to recognize an active and engaged alumni network remains the best way to garner incremental attention or gain access to the “in person” process in spite of the aforementioned technological obstacles.
Wash U as others have referenced (including OP) is underrepresented in some fields but highly respected. I don’t think it should be of a major concern to OP but I don’t think the concern should be dismissed out of hand depending on the eventual career path.