A non-point. Essentially no one lives in society without any property whatsoever, first of all. If you have any income at all, that income represents property automatically: currency. The very constitution ensures that people are to be secure in their property: "...nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation." The people with less property have the ability to acquire more property, as everyone does. Ever heard of Locke's "life, liberty and property"? That is the philosophical underpinning of our constitution. People are benefited greatly (whether you want to call it "directly" or not) simply because defense of property is vital to a civil society as I've explained.
Protection of property rights is not a conflict of interest. You are making no sense on that point. Zero. A conflict of interest involves violating a primary interest in the service of some secondary (illegitimate) interest. In this context means that you have reason to lobby or act to take something from someone else (in this case by voting it into effect) against their interests and only in favor of yours (to which you have no right). Upholding the constitution and rule of law does not in any way fit that scenario since the constitution itself explicitly lists the rights in question and ensuring property rights in general is in the best interest of everyone. Violating property rights actually comes from a conflict of interest, not the reverse. Since the primary purpose of our system of laws and police powers is to uphold the rule of law as constrained by the constitution, doing what is in the constitution is by definition NOT a conflict of interest within the context of the political system. Overturning constitutional protections in service of OTHER interests conflicts with the primary interests of individual rights and a just society.
Welfare can be seen as a conflict of interest because it is an example where the primary interest of upholding the constitution with its bill of rights (including property rights) is in conflict with the secondary interest of getting free shit without having to work for or produce it yourself. If you can just vote yourself into possession of all the creature comforts that you could want at the expense of others and their rights, what in your world view prevents that or makes it illegitimate?
Those are things that you are for (and are your opinions) but to what extent? When does it go too far and when should protections of individual rights be taken into account beyond the crude math of majoritarianism? What if you, along with a 51% majority of the population, could vote the property rights of the other 49% of the population away entirely and have the proceeds go into your own pockets? Or if 51% of the population voted themselves to be perpetually taken care of through the efforts of the other 49% without any expectation that they produce or contribute in any way (which would essentially be slavery)? That would only arise from a conflict of interest and is the most extreme version of what is being spoken of. Hell, what if 51% of the population voted that the rest of the population didn't deserve protection from violence or should be deprived of life or liberty, not just property? Is protecting people from violent crime also a "conflict of interest" in your world? Voting is supposed to be about fostering a society in which individuals can prosper, not allowing groups to prosper simply by taking from others. At what point is voting to take resources away from another group no longer legitimate in your mind?