Can you convince me that pure marketing is a better career path than brand management?

Both have equal importance. It is upto you to choose the field as per your interest and inclination towards the field. 

In my view, marketing is actively promoting a product or service. Marketing is a push tactic to get sales results and this is not branding.

Branding is different than marketing. Branding is not push, but it's a pull tactic. A brand will help encourage someone to buy a product, and it directly supports sales & marketing activities. 

Branding is strategic whereas Marketing is tactical.

Marketing unearths and activates buyers.

Branding makes loyal customers, advocates, even evangelists, out of those who buy.

I think brand management and creation of brand promoters is the most comprehensive career path. 

• 1st u create awareness about a product through the marketing strategy , u select  a media ( online is better if the audience is young. 
• You create a sales strategy that include direct selling. 
• U provide after sales service. it here that customer service comes into force. U convert detractors and fence sitters into promoters. For that u need to have full control over the customer service reps so that they are polite and knowledgeable. Also u need to support products even if they are OUT OF WARRANTY so that the customer remembers u when he makes the next purchase. 
• Support forums chat and email support are inexpensive means to support the customers of oow products. 

They are two pyramids in a full operation, Branding (Product development) and Marketing.

As for the career issue nobody can convince you, it's your call, your choice.       Thank You.