"It has fallen out sometimes that both Papists and Protestants, Jews and Turks, may be embarked on one ship... All the liberty of conscience that ever I pleaded for turns upon these two hinges: that none of these Papists, Protestants, Jews, or Turks be forced to come to the ship's private prayers or worship, nor compelled from their own particular prayers or worship, if they practice any." (Roger Williams, 1654)
In addition to the fact, not noted by Williams, that by offering prayer in public contexts like legislative assemblies, governments or other organizations attempt to put a religious patina upon their work, I find this quote, and the general drift of Williams' thought (who believed in the separation of Church and State for religious reason), amenable.
Prayers in such contexts tend to be generally deistic in nature. Now, if you happen to be a deist, such prayers work well for you. Additionally, if you are of a religious faith for which deistic prayers can be, by mental agility, conformable to your own prayer practices, then it can also work for you.
However, if you are an atheist, or adhere to a non-deistic faith tradition (of which there are many), then such invocations prior to public assemblies force your participation, or at least your presence (and so implicit participation). So, as a Christian, I advocate for an end to such prayers.
We definitely pray for our government and elected leaders, vigorously, and often, in the context in which such prayers are appropriately lifted--Christian worship, or the worship tradition of which you, dear reader, are a part.
Finally, it is not clear to me, outside of the worship life of a specific community, what such prayers at a legislative assembly or other public event (such as the inauguration of a president) might mean. If they are not prayers in the name of a specific deity, to whom are they addressed. It is a dangerous practice to invoke a deity in which you do not believe or trust. They very well may show up. Religious communities typically take great care to identify which god to which they are praying, and include in such prayers specifications as to what this god is like. Again, this makes the most sense within the worship life of specific communities, not generalized and supposedly secular contexts.