He didn't care about the common man? Saving the huge majority of his city from nuclear destruction makes it seem like he cared a pretty decent deal. In any case, alerting the public to odds of a war would've hastened their demise if anything; unless he was in with Vault-Tec and had control over the Vault system, he wouldn't be able t fill the Vaults to proper capacity come October 22nd. Apart from overstepping his boundaries and trying to save all of America, which frankly wouldn't have much sense, he saves what is most accessible to him. Your notion of "saving the common man" needs to have a concise goal around it -- he saved most of his city and likely would've helped the post-War inhabitants if he wasn't taken offline by the blasts.

There are decrepit factories and resources laying around Vegas. You're right that they would be hard to get back into full gear, but they are definitely there and waiting to be utilized by a proper power. If he lacks something, he could do what leaders of civilizations have done for thousands of years and trade with other factions -- that huge supply of caps he's sitting on would likely be good for trading with the NCR. The Enclave didn't have access to leftover industrial plants, nor did they actively trade with the other wasteland factions, nor were they even interested in rebuilding the country unless you count wiping out a good portion of the human race as rebuilding. They were in the presence of a lot of powerful military technology, but we aren't shown they have any industrial capability to rebuild.

To the next point, let's think logically about House and his intentions. He has prepared for a nuclear war long before the Great War itself. He has funneled countless millions of dollars and caps into protecting most of the city from external threat and harm. He has a clear and concise plan with the Platinum Chip and with what the other factions are likely going to do to Vegas. I'm not sure why he would go through all of that to rule from the Lucky 38 over what would be a shithole in the desert if he hadn't improved it. He will rule over it with despotic and autocratic power, yeah, that's something he himself says. He also says he won't interfere with his citizen's personal lives, he will get industrial and medicinal sectors back up in the next few decades and ultimately create a city that will dwarf any other in that half of the country in terms of advancement and living standards. Given that we do know he has resources and cash available to him, why would he go back on his words and his actions? Is there anything ingame or in lore that hints this?

Droughts and natural disasters are challenges, true, but it wouldn't be hard to get an irrigation system up with the influx of caps coming in. Security? Several hundred of the robots you said could easily wipe out a massed army of the Legion and are easily enough to defend Vegas. Given how there are extensive facilities around the Mojave for robotic production, he could bolster his numbers higher. This is what's being left out -- there are a lot of resources around the Mojave, the key is that people either aren't interested in them or they lack the knowledge of using them. You would need something with the power and intelligence of House to get them running again.