Is the halting problem claim true with the advent of AI
Even if we can design better AI solving problems we did not think possible even 10 years ago, they will always be computer programs. And as you said it yourself, "Alan Turing states that, there can't be a program that can decide if a program will ever stop." (he actually proved it and not only stated it...). Thus even the best AI you can think of will not be able to decide this problem, this is uncomputable.

This is a theorem, and unless we find a new device doing computation beyond the power of Turing machine (I am not talking about faster computation, but really a new kind of computing), this will remain impossible. Even if we have whole new kind of computer defining a new concept of computation, let's call them hypercomputers, it is very likely that those hypercomputers will not be able to decide 
if a hypercomputer ever stops.

However, you are right in the sense that in every day life, many programs are simple enough to be analyzed and proven to terminate. Thus we can still hope to have better algorithm (using for example advances in AI), that can generate proof of terminations for a large class of programs. See for example Microsoft Terminator [1] which precisely address this problem in practice.

[1] https://en.wikipedia.org/wiki/Microsoft_Terminator