Thermodynamics has simple answers to offer regarding reversibility or otherwise of a given process. For a process to be reversible, it must be reversible at every point along the path ie, the system must be in equilibrium, both internally having well defined values for its properties such as temperature, pressure , internal energy etc., and externally with the surroundings as well.
In the given process the gas undergoes a complicated process of expansion. It does not satisfy the requirements of reversibility - the system being in equilibrium at every point along the path. Hence the process as a whole is deemed to be irreversible. Acceleration/decceleration being zero is no criterion for reversibility of the process. Slower and slower decceleration rates of expansion, too, do not ensure reversibility of a process. Hence, the process as a whole is irreversible.
It is important to note thermodynamics does not care for the rate(s) at which a process occurs. Therefore, all the elaboration regarding the spherical shape of the container and its instantanious radial expansion followed by a decceleration ending up with a slow steady rate of expansion is of no concern to decide whether the given process is spontaneous (irreversible) or not. Free adiabatic expansion of a gas (whether ideal or not ie whether viscous forces exist or not - assumption of ideal gas simplifies the arguments) is irreversible. Therefore, the answer to the given question is that the process is irreversible. Even the tail process, when the external pressure is infinitesimally small, unless the pressure of the gas is equally infinitesimally small, the process continues to be irreversible! Infact, we can have the expansion going on at a very rapid rate with gas pressure being equal to extenal pressure, the process is deemed to be reversible for an ideal gas!
Given the initial and final equilibrium states A,B of a system, thermodynamics tells us whether it is possible or impossible for the system to go from the A to B, on its own. One criteon used for arring at such a result is to see the change in the entropy of the universe. If it is positive, then we say the process A to B is spontaneous, if negative then the process is not spontaneous (does not occur on its own); if zero the process A to B is said to be reversible.