From Elongated Triangle to Kite-shape shield
I cannot find any in-universe reason as to why this occurred, but out-of-universe I can.
Apparently the original sketch of Captain America came to the attention of Archie Comics (then MLJ Magazines) - they noticed there was a similarity between Captain America's shield and the chest detail of their own character The Shield (below).
Consequently, Captain America's shield was changed to the Kite design.
(Source: Cronin, Brian (2009). Was Superman a Spy?: And Other Comic Book Legends Revealed. Plume. pp. 135–136. ISBN 978-0-452-29532-2.)
From Kite shape shield to Round shield
The Marvel Wikia site refers to President F D Roosevelt presenting Captain America with this. This puts it between 1933 and 1945
From Captain America Vol 1 255: 'The Living Legend' in which
It details the origin of Captain America and focuses on his early
adventures in 1941-42.
As correctly pointed out in the question:
Rogers was originally issued a traditionally "kite" shaped shield made
of mundane steel, as well as a sidearm.
This new disc-shaped shield, however, had two main advantages:
1: It was better as an offensive weapon
that its excellent aerodynamic properties made it an
effective offensive weapon
2: It was made of a nearly indestructible alloy
Originally, Dr. Myron MacLain was attempting to perfect an
indestructible alloy as strong as the legendary 'Adamantine', which
Hercules' Golden Mace was made from. Dr. MacLain was hoping the alloy
would provide a distinct advantage in the armor of American war
machines, during World War II. The Doctor worked tirelessly, using the
exotic Vibranium metal and an unknown substance. While the Doctor was
asleep, as a result of his exhaustion, an unknown factor caused the
metals he was working with to bond.
An almost indestructible shield has obvious advantages over one made from 'mundane steel'!
Not only is the shield nearly indestructible though, it also has other useful properties which are inherent to Wakandan Vibranium, namely the ability to absorb vibratory energies in the vicinity.
The Wiki page on his shield explains the following (however bear in mind that it does not reference the said sources):
Some sources say that Dr. MacLain chose the discus shape because of
its versatility, while other say that he used an existing cast
designed for tank hatches.