Yes, if a plane were to fly through the beam it would break connection
Depending on how long its out of line of sight the error correcting codes might be able to let you ride through it, worst that will happen is you lose the connection entirely and have to start the acquisition process again.
The odds of such an interruption by a plane are vanishingly small but clouds are an issue. We need cloud free line of sight. This is where RF does have an advantage- clouds are not a serious issue for RF. We solve that by putting multiple ground terminals in different geographic locations that are not correlated in weather. So if one is clouded out then we have others to use.