I have one child that clung to me, freaked out, etc.. ! She had meltdowns like you are describing, too. It has been quite a road...Luckily, I have a certification to work w/ families. In some cases, some kids need get out of the habit of treating their parents like doormats and have rules established...Like Supernanny. In other cases, kids are fearful because they are "crying" out for help. One of my kids was crying out for help !!!
Your son is 4 1/2. Is he in preschool? How does he do there? What other things does your son do? Is he sensitive to lights, loud sounds, touch, etc... How are his sleeping habits, eating and so on?
Does he follow a routine at home? Does he go to bed easily or does he struggle to get himself to sleep?
For all the years I have worked w/ families, I have found that a child w/ issues such as yours needs to be seen by more than just a pediatrician.
Sometimes these can be sensory processing issues, anxiety, etc, etc, etc...
Since I did work w/ families and knew what to do...I brought my child to a neuropsych, an OT certified in sensory intergration dysfunction, vision therapy doctor,& a therapist, etc...
When school time came, I fought hard to have her under a law so I can have meetings w/ the school team, when needed !! I found that the teachers were not trained to help my child all that much, so I had to be the advocate because they labeled her slow. That was wrong, so, I had the experts come into the school to talk at the meetings ! If kids like ours do not act out in school--and some don't---then, you will have to be your child's advocate.
Start at the pediatrician...
There is a great book entitled, "The Spirited Child."
Up above, you said that this is about your son's temperament and anxiety. I am wondering how your son does at school, at home...I did see that you cannot sign him up for any sport like activities.
Is your son an introvert or does he truly have heightened fears, etc?...Please get him seen. I would not want him to have to go to Kindergarten and freak out when he has to go to gym class, or let his fears get in the way of his learning, etc...
Take the next few months to get him seen by somebody! Maybe it is just something simple. If it's not, you should really find out !
As a parenting teacher, I can tell you that if a child's fears last for a long period, say, 6 months or more or the fears affect make his daily life activities a struggle or affect his capacity to make friends... Then, the child is "asking" for help and it is time to seek out help from a therapist.