What happens if I only want the brand and customer list, not the physical operations?

That's a legitimate, narrow form of asset purchase — you can structure a deal limited to intellectual property like the trademark, trade name, and goodwill, along with the customer list, while leaving behind equipment, leases, and staff entirely. The seller keeps or winds down the physical operation; you get the brand and the relationships behind it.

The nuance is tax treatment. A joint election is available that lets the parties agree GST/HST won't apply to a qualifying business sale, but it generally requires the buyer to acquire all or substantially all of the property needed to actually carry on the business, not just its name and customer list. A purchase this narrow may not qualify, meaning GST/HST could apply to this transaction where it wouldn't to a sale of the whole going concern — that's worth confirming with an accountant before you price the deal.

It's also worth thinking through what a customer list is actually worth without the operation behind it — customer relationships often depend on the people and service customers are used to, not just the name. A business lawyer can help structure the purchase agreement around exactly what intellectual property and goodwill you're acquiring, and an accountant can confirm the tax treatment before closing.