Link us to what you bought and post a picture of the steps and landing.

 

I haven't purchased anything yet. I get the idea of drilling through the brick into some solid surface, and that's what I tried on my last house, but the holes went through to voids. I also understand I could build a simple wooden railing system abutting the left edge of the step with posts in the ground but I hate the look.

 

While Tapcons will work in the right conditions, wedge anchors are much stronger. Anything that mounts flush will be a compromise. The best way is to core drill the concrete and bury the iron into anchoring concrete. Will the upper section mount to the wall? That helps a lot.

 

How to Avoid The Most Common Problems When Anchoring Metal Railings Into Stone and Concrete Steps - Wiemann Metalcraft

The very first part of a building that we may touch as visitors is the handrailing, even before the door hardware. For the Architect or Builder or Stone Mason, upon return to a former project site, the shock of seeing a railing like this, literally tearing apart the masonry fabric they so...

wmcraft.com

Hope that helps. Common sense says railing will not hold in bricks. You will have to drill into base under the bricks but most such stairs use blocks. You may have to rebuild parts of the stairs, filling the core of the blocks. I would not use small screws. I would have looked for railing system without the base plate for screws. Rather, just longer post so it can be sunk into solid base.
If just 3 steps, seems post doesn't need to be in the tread. Return what you have and buy one with post that can be anchored to the landing. Maybe custom made. Landing may be easier to rebuild, reinforce, already strong enough if it is concrete pad.

 

Anchoring to brick will work if it’s done right, especially if the upper end can mount to the wall. Here’s one I did 10 years ago and still holding up fine. It’s mounted to the block below and brick laid around it

 

You need an 8” L shaped piece on top, then bury the lower post in concrete, that would last a long time. Use the wall on the left to mount the railing, and you will need one surface mount to the brick pavers, guarantee that will hold.

 

I may be repeating racinmason. Same idea anchoring the top rail to left of photo wall with L. But pour a footing next to the steps and anchor a post there. Bottom of the rail also turned. You can't leave a gap between the balusters and the edge of the stairs.

 

When in doubt just add more anchors.