No not really. While they have high fertility, they also have cultural practices that limit the spread of their viewpoints (requirements to live in particular geographies, etc).
I would more be talking about groups like traditional Catholics which have no such prohibitions, and frequently find themselves in very high-ranking fields. For example, I'd point to many Supreme Court Justices, several entrepreneurs, several media figures, etc. It's not just traditional Catholics [1], there are many groups with persistently high fertility rates.
It's not just Catholics. Conservative protestants also have much higher fertility. In particular, red states have higher fertility than blue states, and if you drill down deeper, it's simply due to these states having more of these groups than blue states, not any particular policy. The data are pretty unequivocal that government policy has basically no influence on people's decisions to have children.
Anyway, the higher fertility combined with the ability of full civic / economic participation means these groups will eventually come to dominate. There is simply no other way. We can talk about immigration until the cows come home, but overall, the global birth rate is sub replacement, so while any one country can migrate more people to solve its problem, this just punts the political football to another country which will invariably become more conservative.
To conclude, I can name several examples of 'conservative' groups with significantly higher birth rates, but I can't name a single liberal one. This is simply a cultural thing, but it doesn't matter for the economy. Regardless of who has kids, the kids pay the taxes necessary to sustain the welfare state. However, as I pointed out, it'll likely lead to political differences that many here won't like. As an example, I'd point to everyone's surprise that Gen Z is more conservative. I've been saying this for years now, because I first noticed the differential in fertility rates as a child in the 90s. People thought I was a chicken little, but we're seeing the results.
[1] Note, I'm calling out traditional Catholics here. The overall fertility rate for Catholics is on par with an average American. However, simply by breeding, there's already a huge faction within the Catholic church writ large. The progressives, with fewer kids, are dying out to be replaced by younger, more zealous figures. I think this will probably be the next step for the country as a whole as well
[2] EDIT: someone mentioned mormons... that's another good point. A small group of a few dozens of families moving west and now dominate one state and are important minorities in several others.