The biggest issue IMO is lack of flexibility. Leetcode is a specific format with specific questions, but is not necessarily the work you'll do day-to-day.
What would be great is if companies offered multiple formats including leetcode, take home, pair programming, Github review, project review, just talking, etc.
There are several issues that leetcode solves that the other ones don't work as well with or have flaws that prevent their use in some hiring processes.
Leetcode has a single rubric. The same questions and the same answers. Conducting it has minimal additional effort on the employer's side. It does have false negatives, but it has fewer false positives.
Take homes are often similar to leetcode, but some less scrupulous companies have used them for unpaid work (note that this becomes more and more of an issue as the take home becomes more and more relevant to hiring company) the and unless the interviewer has done the take home themselves to benchmark it (this should take between 1-2 hours), it is possible that the request be "do this thing that will take 20 hours."
GitHub review requires that the person have a GitHub project of sufficient size, that it isn't just copied from someone else and that they have time to work on it. There's a joke in /r/adventofcode that shows a GH repo that only has activity in December.
With pair programming, project review, and just talking the lack of a rubic becomes an issue and it becomes more likely that biases get into the interviewing process and leads to possible issues of discrimination.
When mixing any of these so that some people do leetcode, others do pair programming then it becomes an issue that not everyone is considered using the same scale and those biases and discrimination issues become even more something that legal starts wondering about if they'll have to deal with a lawsuit.
If one person can do the interview with "just talking", then that should be the same criteria that everyone uses (no leetcode) and "just talking" has an abysmal false positive rate.
What would be great is if companies offered multiple formats including leetcode, take home, pair programming, Github review, project review, just talking, etc.