I don't think Alexei was spoiled in a way that negatively affected his character. He was certainly coddled by his mother and doted on by his sisters, but this doesn't seem to have ruined his character or made him insensitive.
I think that Alexei's spoiling
did affect him negatively and caused him to be rather insensitive at times, especially in his younger years. There is an instance described in
The Fate of the Romanovs where an eyewitness who lived near Livadia described how the heir “liked to greet people who bowed to him with a bloody nose by hitting them in the face as they bowed,” and when he was not allowed to do that, he greeted them with “very bad language” instead.
Firstly, I cannot fault Alexei for this behavior, because what else would you expect from a child who felt so exalted and indulged from his earliest years, and who was rarely punished or disciplined. Secondly, and
fortunately, I know that he eventually outgrew this bad behavior and empathized more with people as he matured and as his own sufferings increased.
He also began to display quite a firm little character which showed promise that perhaps he would have had a stronger personality than his father. However, it really is impossible to tell what would have happened with him, given his physical limitations. I think he would have had to have a very strong character indeed to make people really respect him and not think of him as an invalid.