I would say that monuments seldom ever have a significant impact on the start of the game and even in the late game they usually only make a difference with niche strategies built around particular monuments. Achieving synergy with regards to tags, religions, and cultures to make monuments both effective and worthwhile is very difficult.
In terms of the 1444 start, not all monuments start constructed or at a level that is significant. Of those that do, a minority are actually useful and almost none affect the trajectory of the game. Some Italian tags like Venice and Florence get minor bonuses that they would not get outside of Leviathan and they're probably the most dramatic wonders in 1444.
Florence gets -1% prestige decay, +1 splendor, and -10% advisor cost from its starting monument.
Venice gets +0.25 yearly republican tradition, +20% reform progress growth, and +0.5 yearly prestige from its starting monument.
The Byzantines have arguably the most useful early game monument in the Hagia Sophia, which grants minor boosts to patriarch authority, clergy loyalty, and tolerance of both heretics and the true faith.
Overall, these are very minor and will not substantially affect your gameplay experience. I mostly view wonders as trophies and money/manpower dumps if I'm not aggressively min-maxing. Some players (like the British) will cross the world to get a wonder that boosts naval forcelimit or something, but generally wonders never grant a net positive return on investment and are mostly a flavor thing. There are exceptions, but it's not a big deal in my opinion.
