Your question shows you are knowledgeable. You give too few facts. An attorney needs to review your questions with more factual information.
You don't say what type of business, and an "associate" may or may not be able to lawfully continue operations under your dad's license. . Some licenses, such as a real estate broker's license, require that another sponsoring broker step in to obtain new listing agreements and buyer's representation agreements upon the death of the broker (even if the associate is also a broker).
What the associate takes at your dad's death may be governed by the organizational documents of the entity. You do not describe the entity type, or the associate's interest.
Proving up a handwritten or holographic will is not always easy.
A lawyer can't answer your questions without more information. It often takes far longer than 120 days for probate to be completed by the executor.
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You need to take the matter to a probate attorney. Oftentimes, a business can be run following the death of the primary management. One should look to the business' organizational documents for assistance. In regards to actual ownership of the company, one would probably look to probate. Remember, ownership and management are two different things.
The above statements are provided as general information and not intended as legal advice. Each matter has its own set of unique circumstances that cannot be adequately addressed without consultation. You are strongly advised to hire an attorney licensed to practice law in your state to represent you.
In addition to the previous answers, I do not think that the shares pass automatically (but would require a corporate resolution as well as the personal representative of the estate's approval). Once a personal representative is appointed to the Estate (you might need a Temporary Administrator if the holographic will does not name an executor), the Temporary Administrator could seek Court permission to operate and exercise your late father's interest or ask that the court appoint a receiver for those interests.
Texas Probate Code Sec. 238 provides guidance to a dependent administrator (be it a business, farm, ranch, or factory) in this situation.