Family-owned companies often (not always) have a very strong sense of personal enmeshment in their company. They often use language like “my baby” to refer to their business. Imagine a protective father whose daughter is being proposed to. It often helps if the owner feels that the buyer respects their company, its values, the staff, their customers and so on.
I would have a good conversation, perhaps several, with the owner(s). Also, for family-owned SME’s, while one owner may have a controlling stake in the company there may be someone else (their spouse, a divested founder, a close advisor) who may still have a say in decision-making. Make sure you know who the decision-makers are, their decision-making process, and what their interests are.
If they come to feel they can trust you, that you care about their company and people and that you have the ability to run the company effectively – you will greatly improve the likelihood of having your offer received.