It looks like the other "weed" is prostrate spurge.

Question still remains: Are there any reasons I should not encourage these "weeds" to grow as groundcover?

Thanks

prostrate spurge has a central root and as such may not control run off of moisture. Now that you have it you probably have it for life. I do like the leaves and it does a a function to cover bare ground until I decide what I want to put there. Some will throw up in horror but it is a nice weed.

And purslane is good to eat. However, both germinate VERY late in the season, at least in my neck of the woods, so may not keep out other weeds very well.

I think the purslane is an annual so other plants might fill the space when it is not in season. Around here it seems to inhabit cracks in pavement during the summer.

There can be Purslane, Chickweed, prostrate spurge, Black Medic, and some others that grow like that that most people consider "weeds", but like any other plant they do have their place in our world, just not in my yard. Whether your neighbors would appreciate you allowing these plants to grow in your yard depends on how anal they are about their yards. Many places are putting in place ordinances that state they you cannot have any vegetative growth in excess of 8 inches, but do not address these types of "weeds".