If I am picturing the correct product this edge is porous to allow water to flow through it. That being said there is still a chance grass may poke through. If it does is will not be to the degree you are dealing with now. As for a brick border, this will depend on what is between your pavers. If you use a polymeric sand (sand that contains a "glue") it will bind the sand particles together when you moisten it so it will help prevent weed growth between the pavers. This also eliminates having to re-apply the sand annually as is often the case with a regular sweep sand.

Just keep in mind if you use a rubber mulch edging it may not blend well with the mulch or other material you have inside your planting bed and may stick out like a sore thumb.

please do not use the rubber mulch, it is toxic,google rubber mulch and find out for yourself.

I don't have experience with that product but can tell you from experience of keeping my yard that I haven't seen any edging that will prevent grass from growing up and over it.

I have brick edging and concrete along the driveway and I have to edge it once a month to keep it looking nice.
Around the flower bed I use a weedeater but along the brick/concrete I use a gas edger. Even tho my DH can mow right over the top edge of the brick, he can't mow close enough to keep the grass off it.

If your DH won't trim the grass, then buy a trimmer that you're comfortable using. If my 90 yr old mother can use one, I'm sure you can to! :)