If the will is valid, the named executor is the executor
Technically, the person appointed by the court is an administrator, not an executor. The correct course of action is for them to notify the court and ask for probate to be granted to the named executor to replace the letters of administration.
An executor is a trustee
They have a fiduciary duty to act in the best interest of the beneficiaries. How they choose to discharge that obligation is up to them. They are certainly not required to have witnesses to their activities but if there is a lack of trust in the trust a prudent executor may want a witness for their own protection.