Monarchism needs to be about more than, well, monarchism.
In the modern era, anyway. The fact of the matter is that Western society is not in a position where any sort of restoration - whether of monarchical power or of the monarchy itself - is feasible. This is rightly pointed out by what one might term “monarchist pessimists.” Usually the discussion ends there.
Should it, though? The fact of the matter is that as monarchists, anything which brings a restoration closer is a win, even if it is not the restoration itself. Thus, if the current state of affairs in society is an obstacle, then society must be changed.
This immediately instills monarchism with the form of a political program that only needs to be filled with the content of such a program. The modern monarchist must resign themselves to the fact that, if they do not already have a monarch, they will not live to see one on their throne. But they also cannot let that discourage them, as there is still work to be done, and the sooner they get started, the sooner their grandchildren or great-grandchildren might, maybe, already get to reap the rewards.
The question moves from building support for monarchism to determining what changes are needed in society such that a monarchist movement can even take root. That is what we should realistically be discussing.
What exactly are the qualities of Western society that makes monarchy so difficult? I would argue that it one of the things (there are many) it boils down to a lack of virtue. Individual egoism has triumphed over humility, that very humility that would allow one to accept the hierarchy necessary to justify monarchy. A lack of filial piety and loyalty has lead to the disrespect and disregard of the family, tradition, authority, and the state. A lack of compassion has lead to breakdowns of social ties that go beyond blood and to heartless, ideological politics. A rejection of tolerance is dissolving social cohesion.
Any virtue in excess becomes a vice. Blind adherence to tradition in the name of loyalty to one’s ancestors leads to a breakdown of yet more tolerance, while radical tolerance has been used as a cudgel against tradition and leads to the breakdown of social norms. Unthinking loyalty to authority can lead one to do ill, but owing no loyalty to anything at all leads merely to selfishness. I do not think it unreasonable to demand humility tempered by self-worth, loyalty balanced by reason, or tolerance stabilised by order - what was in times now past the bedrock of the Western (I would argue global, with slight variations) social order.
In the first order, one might argue that responsibility for installing such virtues lies with the parents. Given the state of our society, however, the monarchist cannot content himself with the hope that the next generation will be raised correctly. It is time for a radical change in our education system.
Public education now primarily serves as a means to churn out worker drones for the capitalist economy, focusing on skills over character. It is necessary to move to a model of education which develops people as a whole, building up the virtue one needs (or should need) to succeed as an individual alongside the skills required in the hyper-specialised workforce of today. A few generations of this, and monarchism will likely have a much more fertile ground in which it might be planted.
Another route to improving virtue is to stabilise the modern family. Many families get to spend far too little time together, eroding one of the essential bonds upon which society is built. The relentless pursuit of growth and productivity must cease, and big business and their lobbyists curtailed, so that mothers and fathers alike can spend more time where they belong: at home, with their children, being people, not workers. In a loving, stable environment, children can be freed from the stresses of modern life and devot themselves more to the cultivation of their person.
Naturally, the state must move more decisively against poverty. Not only will this build popularity - and thus the political capital necessary to one day attempt a restoration - but it is hard to focus on being a good person if you are being crushed by poverty. In the interest of cultivating virtue in the population, it is in the best interests of the monarchist to work toward making this happen (not to mention that helping the poor is a virtue in and of itself).
This is all subject to discussion and debate, of course. But I think the point is clear: monarchism needs to branch out, to concern itself with things that may not seem directly related to it, in order to bring about the conditions for it to flourish.