I'm thinking these could either be their own full block, or a carpet style that would be similar to these guys https://www.brickowl.com/catalog/lego-dark-stone-gray-tile-2-x-2-with-groove-3068

Run speed would be increased while on them rather than dealing damage.

OHHH this is what you meant. I thought you just wanted a Plastic Construction Brick sans toe injuries. This is a pretty cool idea. Perhaps going one step further than the carpet idea, what if they behaved like Snow Layers and could be stacked (just as real legos can be)? So they could work like carpets, but also like paths (take up less than a full block height) depending on how many you stack. It wouldn't change the speed bonus, just be cosmetic for more build versatility

I do quite like the idea of multiple-layer flat blocks... would need another 16 blocks added (for each colour) plus 8 models per block (assuming 2-pixel layers, like snow is). Mixing colour layers within one block would not be an option...

Smooth construction bricks are in 1.18.2 and increase the player's run speed by ~50%. Still thinking about how multi-layer bricks would be done; two main considerations are:

• Need side textures for the bricks (top & bottom not a problem) that would work well for all heights. Just taking texture slices out of the main textures doesn't necessarily work, since the bricks have edge highlights/lowlights. @Ridanisaurus might be able to offer some advice here.
• User interaction: how does a player modify the height of these bricks in a user-friendly way. Repeatedly wrenching a brick to adjust its height 1 step at a time isn't really feasible, especially for large builds...

What does the base game do to make this possible w/ snow layers?

Snow doesn't have edge highlights, it's just a flat texture.

• Need side textures for the bricks (top & bottom not a problem) that would work well for all heights. Just taking texture slices out of the main textures doesn't necessarily work, since the bricks have edge highlights/lowlights. @Ridanisaurus might be able to offer some advice here.

What does the base game do to make this possible w/ snow layers?

• User interaction: how does a player modify the height of these bricks in a user-friendly way. Repeatedly wrenching a brick to adjust its height 1 step at a time isn't really feasible, especially for large builds...

Perhaps some sort of tool that can remember a configured height via left click and deploy a block of that height on right click? Like a plastic construction foam gun or something. Doesn't necesseraly have to be a new tool item, perhaps it can just be a mechanic of the actual block's item stack in the player's inventory or a behavior added to one of the existing tools.

To stay try to the Minecraft looks and feel, the cap can be the same thickness as a slab, but can only be placed on top of a Plastic Block. However, that is a little bit thicker than the Lego Caps, but I think it would be make it easy to fit with builds, at least in my opinion.
Making textures for the top/side/bottom of these should be fairly straight forward. You just let me know what you decide, and I will work on it.

Would it be easier to just treat it as a carpet and/or slab, rather than layerable like snow?

For the interactions, Engineer's Decor has rebar concrete slices that you adjust the height on by right clicking the top (or bottom) with a slice to build up and left click to break down. Maybe that might be a usable approach for this?