Convert recessed brick mortar to concave mortar design to decrease water intrusion due to high winds
There’s no reason a new mortar mix will not adhere and perform adequately if, 1) the joint is cleaned properly, 2) proper mix AND installation is used, 3) infiltration is not caused by settlement 

1) You emphasize that it’s not a repointing issue. That makes me believe you’re not going to clean the existing mortar joint...bad idea. In fact, you have 3 sides of those mortar joints to clean. 

2) Here is a link from Cemex that explains the types of mortar. You’ll want an exterior mortar that can be troweled concave. (Concave helps “squish” the mortar into the block and provides a hard sealed surface to shed the water. As you know, the recessed mortar joint allows water to lay on the brick and wind will “push” it into the building.)

https://www.cemexusa.com/products-and-services/cement/product-specifications/type-n-s-m-masonry-cement-and-mortar

Btw, pay attention to their discussion about when to strike the mortar so you don’t remove the moisture, which will cause cracking, etc.

3) I suspect you have a greater problem than the type of mortar joint. If water is “gushing in” when you squirt the wall, the joints must have significant cracks between the mortar and brick. 

With brick veneer walls, we know they are going to leak...somewhat...that’s why a moisture barrier is installed with weep holes.  

We have removed sections of the interior wall to re-install new studs, plywood, and moisture barrier...along with new ties drilled into the back of the brick veneer. (Btw, it was a two story building.) This is extremely difficult and time consuming, but may be your only course. 

Lastly, I’d do a test section and see how it goes. Give it time to cure and see if it works.