What is the best way to explain to a client that their logo is incredibly poor and amateurish when they seem to like it?
They want to spend a fair bit of time and money developing new products,designs and marketing but their current logo is a real obstacle in making them look professional.
I have tried substituting the logo with a reworked version during demos of new products and explained it as "giving them a new option". They simply preferred the version using their original logo.
I have also been much more blunt. Stating that their logo is looking dated and doesn't reflect their business.
The current logo style has no special relevance to the industry that they are involved in. They are not a major brand where it would cost many thousands to change their branding and merchandise so cost is not a realistic barrier.
To give some sort of an answer for you:
Sales figures on similar branding update data would be one very good approach. Just make sure it relates and had a similar reason. A racist logo from the 1920s updating to not be racist is different then not liking the color choice or something. Data from one would have very little meaning on the other.
If the new marketing you're going to be creating includes physical objects such as shirts or hats that might be expensive with the current number of colors and details that a proposed logo update would solve.
Performing an A/B study showing that a proposed alternative is more memorable to the intended purchaser.
If there was a merger, fundamental shift in corporate philosophy or products, or other big change. One that comes to my mind is Nintendo changed its logo in the 1980s when it switched from trading cards to video game systems.
Here is an article on Forbes about when to change logo:
The last sentence is
Which brings us to the most important point of all: when your logo is not just identified but liked err on the side of leaving it be.
The owners don't appear to have asked you to change the logo and they like it. Unless you can come up with a very valid reason, err on the side of leaving it be.