There are two distinct questions that you might be asking.

1. Why has the mathematical community adopted ZFC as a standard foundation and not ZF+GCH?

2. What mathematical and philosophical arguments can be advanced for and against adopting AC and/or GCH as a fundamental axiom?

Much as we might like to believe that the reason for the sociological acceptance of ZFC is that there are strong rational arguments for it, and that mathematicians accept those strong rational arguments because mathematicians are strongly rational, I do not believe that this is true. I believe that ZFC has been adopted as the standard largely for historical reasons. At some point, the mathematical community recognized the value of having some standard, because it would demonstrate that all of mathematics could in principle be formalized in a single system that avoided all the known paradoxes, and ZFC just happened to show up at the right place at the right time. I suspect that various other candidates could have "landed the job," including ZF + V=L, or even Z + C, and that ZFC was just a bit lucky.

After a choice was made, most of the mathematical community lost interest in foundations and so had no real interest in tinkering with this or that axiom to get "better" foundations. I don't think that ZF+GCH was ever a serious contender, because GCH was still considered an open problem by the time ZFC had already secured its status as "the" foundation. If a statement X is considered an "open problem" then people generally do not also consider X to be a candidate for a basic axiom. By the time the independence of GCH was established, it was too late to apply for the job.

Having said all that, I should add that if by "expert mathematicians" you mean experts in set theory, then the question is a bit different, because set theorists are more interested in these sorts of questions than the mathematical community as a whole is. Some of them will tell you that they reject GCH as an axiom simply because they don't believe that GCH is true. Still, even among set theorists, a sizable proportion take what you might call a "pragmatic" approach. They care mostly about whether the standard base theory is a technically convenient one for the investigations that they are interested in. Then the considerations that Asaf Karagila mentions come into play. GCH just isn't the most natural or convenient axiom to use for most things that set theorists currently care about. If it does happen to be useful in a certain context then they won't hesitate to assume it, but such occasions don't come up that often. (By the way, often Martin's axiom turns out to be what you really need when you might think you need CH.)

There are two distinct questions that you might be asking.

1. Why has the mathematical community adopted ZFC as a standard foundation and not ZF+GCH?

2. What mathematical and philosophical arguments can be advanced for and against adopting AC and/or GCH as a fundamental axiom?

Much as we might like to believe that the reason for the sociological acceptance of ZFC is that there are strong rational arguments for it, and that mathematicians accept those strong rational arguments because mathematicians are strongly rational, I do not believe that this is true. I believe that ZFC has been adopted as the standard largely for historical reasons. At some point, the mathematical community recognized the value of having some standard, because it would demonstrate that all of mathematics could in principle be formalized in a single system that avoided all the known paradoxes, and ZFC just happened to show up at the right place at the right time. I suspect that various other candidates could have "landed the job," including ZF + V=L, or even Z + C, and that ZFC was just a bit lucky.

After a choice was made, most of the mathematical community lost interest in foundations and so had no real interest in tinkering with this or that axiom to get "better" foundations. I don't think that ZF+GCH was ever a serious contender, because GCH was still considered an open problem by the time ZFC had already secured its status as "the" foundation. If a statement X is considered an "open problem" then people generally do not also consider X to be a candidate for a basic axiom. By the time the independence of GCH was established, it was too late to apply for the job.

Having said all that, I should add that if by "expert mathematicians" you mean experts in set theory, then the question is a bit different, because set theorists are more interested in these sorts of questions than the mathematical community as a whole is. Some of them will tell you that they reject GCH as an axiom simply because they don't believe that GCH is true. Still, even among set theorists, a sizable proportion take what you might call a "pragmatic" approach. They care mostly about whether the standard base theory is a technically convenient one for the investigations that they are interested in. Then the considerations that Asaf Karagila mentions come into play. GCH just isn't the most natural or convenient axiom to use for most things that set theorists currently care about. If it does happen to be useful in a certain context then they won't hesitate to assume it, but such occasions don't come up that often. (By the way, often Martin's axiom turns out to be what you really need when you might think you need CH.)