NOTE: Complete amateur at this, this was my first time at this on this scale, so....
3 years ago I laid new tile in our new Kitchen and Breakfast nook as part of a remodel. After some research I went with the recommendation of a high end granite and tile store for the sealer. Miracle brand 511 is what they use.
I used two types...
First I applied the 511 Impregnator. (Designed for medium to dense surfaces)
Then I applied a second coat of the 511 Porous Plus. (Designed for more porous surfaces)
I think either one alone would have done just fine with two coats, but IIRC, it was explained to me that the Impregnator is thinner, made with smaller molecules and is supposed to penetrate deeper and bond with the grout, but it won't necessarily fill and seal off the larger pores. The Porous Plus is supposed to be slightly thicker and tends to fill and seal over any remaining pores better.
Both are rated indoor or outdoor and I hope they hold up.
I applied with a roller tipped applicator that was roughly the width of the grout lines, very similar to your brush applicator. It went pretty slow, my wife went behind me cleaning / soaking up any excess, per the instructions. I did have some trouble with the sealer drying too fast and gumming up the small exit hole in the applicator. My tile doesn't require sealing, just the Grout.
We used a very dark gray grout just for avoiding seeing stains etc so it's very hard to tell if the grout is staying clean and effective or not. Basically we did it and forgot about it, except for making it a point not to use any harsh cleaners that might damage the sealer, like ammonia or bleach.
Collecting dust.