It wouldn't hurt. looks like a lot of that mortar is just missing with holes and cracks. Looks like rain and a bit of wind could drive rain right through it.
That said, it looks like the brick is also damages. Brick with a damaged face will also take in water like a sponge. They do make spray-on brick sealant. I can't say it it works or not, but might be something to look into.
It depends...
These days, bricks are used far less frequently as the "structure" of residential homes, due to the higher chance of failure from lateral movement (such as an earthquake). Depending on the construction of your home, they may provide little to no structural value and may be purely cosmetic. If they are cosmetic, the structure should not be compromised any more than it was when the mortar failed (which would typically mean "insignificant"). If they are structural, your concerns may be warranted.
If your house was built more recently, uses primarily wood framing including along the interior sides of your exterior walls, and your rafters/trusses are fastened to said framing (rather than fastened to the bricks), it is more likely to be cosmetic. It is extremely telling in the case where the brick stops before it reaches your roofing timbers. That is not to say the brick provides absolutely no structural value -- it still serves a purpose and can help resist movement, but cosmetic brick does not bear the structural loads. If you've seen a home built with brick recently, you likely saw a free-standing structure, covered with plywood and a waterproofing/resistant/wicking/other material, before the brick "walls" were added.
One of the biggest differences between cement and mortar is typically the lack of aggregate (sand) in cement. If your mixture contained sand, you probably used the mortar as intended. The sand exists, essentially, to give the cement something to stick to, just like rocks do for concrete. Without the sand, I suspect that the cement will eventually loosen, crack, and fall away much sooner than mortar; it also will not prevent movement of the bricks in the same way mortar does, because the "connection" that the provides is missing.
If you cannot determine that your brick is solely cosmetic, I would recommend that you remove the cement and replace the bricks using mortar. When in doubt, consult a professional about your specific situation.
Best Answer
I think you may have a problem brewing there. I have never heard of bricks being glued in a situation like this. Normally paving bricks will be set on a compacted base of stone dust and very fine packing gravel, then the joints are filled with fine mason sand. Mortar is rarely used, especially in cold climates as it will crack with any movement associated with freezing or heaving.
If your bricks are moving under your feet, then they were not seated properly or the base is not properly leveled or compacted. It is not acceptable to have any movement in the field from foot traffic. If your contractor says that is normal, or won't come back to fix the problem, You have a problem. The glue comment throws me. If your contractor said he "glued" the bricks in place, then I think you may have hired the wrong contractor. Maybe he specializes in hanging wallpaper!