Following your logic, individual contributors working remotely are not software engineers, which is ludicrous.

The thought process or tools they used to reach a solution is not as relevant as the solution itself. Interviewers shouldn't just review syntax, but efficiency, comprehension, algorithms, creativity, and many other attributes the code itself tells you. After all, these are things we spend most of our time evaluating in code reviews.

Communication is also key, which can be evaluated in a live review of the code. Programmers are usually happy to talk about their code, and here you can assess not only if they actually wrote it, but what tradeoffs they made, what difficulties they ran into, how they would further improve it, etc. This is also a chance for the interviewee to evaluate the company's code review process; what they focus on, how _they_ communicate, etc. It's incredibly valuable for both sides.

What live coding assesses most is the ability to think on your feet, and performance under the spotlight. This is an unnatural state for programming, which requires deep and creative thinking, often done best in isolation. Not to mention that it's heavily biased against individuals with any form of performance or social anxiety, which sends all the wrong signals you don't want when evaluating someone's programming abilities.

AI tools are simply an evolution of the software engineer's toolbelt, and if they're allowed on the job itself, they should also be allowed in the take home assignment.

It's a mistake to create fake working conditions for potential hires that don't exist for actual employees. At that point you're evaluating based on some fabricated environment, and won't be certain what it's like to work with this person, which is the only thing that ultimately matters.