From the PHB, page 193:
Sometimes you want to get the jump on a foe or wait for a particular circumstance before you act. To do so, you can take the Ready action on your turn so that you can act later in the round using your reaction.
And page 189:
A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn. The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative. Once everyone has taken a turn, the fight continues to the next round if neither side has defeated the other.
Taken together, these seem to imply that, by RAW, when you ready an action, you can only use that action before the end of the round. Once the last combatant in the initiative order finishes their turn and the next round begins, if you haven't used your reaction yet, then you're out of luck.
This essentially means that, if you are last or close to last in initiative order, you are effectively barred from using readied actions for that entire combat. This can create some strange situations.
For example, I recently had the following happen: Our party took some goblins by surprise. Only one member (the scout) was close enough to attack them, so the rest of us chose to stay back and instead ready actions to attack at range as soon as our scout flushed out the enemy. Combat begins, initiative is rolled, and we take our surprise round. The scout moves forwards and attacks one goblin, while the rest of us use ready actions.
On the next round, the goblins, seeing that they are outnumbered, begin to flee. One of them runs for help, and my two allies use their reaction to attack, but when I try to attack as well, I'm told by the DM that I can't.
Why? Well, because I rolled a low initiative. My two allies rolled high, and thus were able to act before the goblins moved, readying their actions during the first round of combat. I, on the other hand, hadn't yet acted during that round, and since the round had ended between my first ready action (during the surprise round) and now, it was no longer valid.
So, my questions are twofold. First, is this how the rules actually work as written? Are my DM and I reading this correctly? Or is there a passage somewhere else that we're missing?
Second, assuming the answer to #1 is yes, would it break anything to allow readied actions to persist until the beginning of your next turn, rather than the end of the current round? How so?