Yes, your website can affect how Google evaluates your Local Services Ads business, but it is not the only factor that determines visibility. Google also looks at things like your business profile, reviews, proximity to the customer, service areas, responsiveness, and competition.
Your website should clearly explain what services you offer, where you operate, and how customers can contact you. Your business name, phone number, and other details should also be accurate and consistent with your LSA and Google Business Profile information.
That said, simply having a better website does not guarantee higher LSA rankings. Think of your website as one part of the overall business profile rather than the main driver of LSA visibility.