It's possible that they will become less biased in their admissions, but it's also possible they'll become more biased.
The University of California has also announced that they won't be considering SAT/ACT scores for admission. California forbids affirmative action in their state schools and while the SAT/ACT aren't objective tests, people often treat them as such and it provides a number for people to compare (whereas you can't really compare "really good at guitar" with "volunteers planting trees"). It's led to Berkeley being 2% Black and UCLA being 3% Black while Stanford is 7% Black and USC is 5% Black. Also notable, Berkeley is 24% White and UCLA is 26% White while Stanford is 32% White and USC is 37% White; Berkeley/UCLA are 36%/28% Asian vs 23%/21% for Stanford/USC. The University of California system is certainly having its admissions decisions impacted by California laws against affirmative action in their public universities.
I don't expect Harvard's admissions makeup to change significantly. I just expect it to be harder for third-parties (who might want to sue) to compare data. If you're qualitatively comparing applications that take a lot of time to review, it makes a lawsuit very difficult. If you're able to compare a numeric score, it makes it less time-consuming (even if the numeric score is biased/inaccurate, people often treat it as unbiased/accurate). It's easier to argue qualitative differences in judgement too and means that you have to start challenging things individually rather than as a class (which basically makes it impossible).
With the University of California, it will be interesting to see if they start coming closer to peer institutions in their student body.