Let's first divide this issue into two different parts:
1.ADD OR REMOVE A MEMBER and ADD OWNER
2.SEND AS, SEND ON BEHALF and SHARED MAIL BOX FULL ACCESS
The first part is the default permissions which an Exchange admin should have.
Besides, these permissions will also be available if the user is the owner of this group.
If as an Exchange admin you cannot manage the group membership, please have a check in Exchange Admin Center>Roles>Admin Roles to see if the Distribution Group role is missing from the Organization Management Group.
The second part is the permissions that are configured on the specific mailbox.
It does not matter if you are an Exchange admin or not.
(In other words admins cannot SendAs/SendOnBehalf or Full Access other mailboxes if they are not assigned these permissions to the mailboxes)
Please refer to this link to have a check if you have assigned the delegate permission of the mailbox:
Manage permissions for recipients in Exchange Online
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