I think the answer to this really depends on the instrument and what level programs you are applying for. While it is really inequitable, it is just fact that a large percentage of students auditioning for the top conservatories are going to be recording in nice halls with a good recording engineer. (Incidentally, it is actually better for sound to record in a hall than a lot of studios since you cannot do any editing of the sound quality for audition videos.) This is especially true for popular instruments like violin, cello, and piano.
Yes, teachers can tell if you are competent or not regardless of the video quality. But if you are borderline, especially for a really competitive program or on a competitive instrument, a better recording might just give your kid the edge. There are minute differences between the top candidates at some of these schools.
If you are applying to more middle-level programs or you have no concerns about your child passing the prescreens at the schools they are applying to, then it is less of an issue, though I would still make sure to invest in a decent microphone.
Not going to lie – my son only applied to 5 schools, but the requirements varied a lot, so he required 4 recording sessions to record his 7 pieces. It ended up costing us over $1000 between the recording engineer and the hall, which luckily we were able to get partially funded by a scholarship. But the investment ended up being very worth it in his case, as the amount of scholarships he ended up getting dramatically eclipsed the amount of money we spent on recordings.
Also look into what free options might be available to you for recording. One of my son’s programs lends recording equipment, and our city has free youth recording studios with quality sound equipment. Churches will often let you record there in exchange for playing at a service (no matter your religious background). Programs will often have halls you can use for free, and sometimes colleges will as well.