I'm not saying that Gwanggaeto's death before 413 would have changed his determination to conquer Baekje; I'm just saying that Baekje might have seemed more of a nuisance because they had allied with Japan (even though they did so in OTL also, I'm just saying that this was clearly possible), which was absolutely unacceptable to him considering that both Goguryeo and Baekje were descended from the same royal blood and that Baekje did not need 'barbarian' help.
Did he care?
Did Gwanggaeto have a determination to conquer Pekche? Did he have the means?
Jangsu did not. He accepted Pekche as a vassal state after 413. As had Gwanggaeto between 400 and 413.
Now, OTL, problems arose during the reign of King Gaero. He seized Koguryo city of Kaesong by surprise attack and held it. He tried to ally with Northern Wei against Koguryo, but Wei was busy on other fronts.
Jangsu made a long preparation for war. In 475, he attacked and conquered the longtime Pekche capital of Seoul. King Gaero was killed.
However, even then was Jangsu unable to annex Pekche. The crown prince held out in southern Pekche; and although the people of Pekche fought against each other and two Pekche kings were killed in 4 years, Pekche founded a new capital in Gongju. The Pekche people did not submit to Koguryo.
Just because the Yan had a dynasty change and the king had connections with the Goguryeo royal family would not necessarily mean that he would cease attacks. It just meant that Goguryeo was expanding its influence and therefore had an excuse to attack the country because it was obviously culturally influenced by Goguryeo.
No. Gwanggaeto treated it as an excuse not to attack, rather than an excuse to attack.
And Koguryo was culturally influenced by China, not vice versa.
Again, this is alternate history. I'm not saying that anything that he did before his extensive conquests would have changed the fact that he died of an illness, probably from overextension, in 413. I'm just saying that if he had lived longer, then he would probably have gone far as to invade China and unite it while the divided kingdoms were squabbling for control.
Since Jangsu did not, Gwanggaeto probably did not have the opportunity.
But now consider the next chance.
OTL in 427, Jangsu moved the Koguryo capital from the upper Yalu valley to Pyongyang.
Gwanggaeto had made Liao river the western border of Koguryo against Northern Yan. The Northern Yan capital was in the Liaoxi region, in Jinzhou.
After 431, Northern Wei attacked Northern Yan. The defences of Northern Yan started to crumble. Generals surrendered to Wei with their fortresses.
In 436, Northern Wei was about to complete the conquest of Northern Yan.
It was then that a Koguryo army came to assist the Yan capital. But they did not even plan to defend the city. Instead they looted the city and escorted the emperor Feng Hong and the people to Koguryo borders beyond Liao River.
Many Northern Yan people did not want to leave their homes and evacuate, preferring to submit to Northern Wei. Some actually mutinied to help the Northern Wei, but were killed.
But Northern Yan ruler though that he was still the legitimate emperor, and that Koguryo ruler was just his subordinate. He wanted to order Jangsu and his officials around.
Jangsu had initially refused to extradite Feng Hong on the demand of Northern Wei. But after two year, in 438, Jangsu was so angry with Feng Hong that he ordered Feng Hong killed, and did not even want to accept the request of their common ally Liu Song that Feng Hong should be sent there instead.
Now imagine the timeline...
Koguryo makes peace with Northern Yan in 408 as per OTL.
But the capital is NOT moved to Pyongyang in 427. Initially it remains in upper Yalu valley.
After 432, Koguryo sends a large army to defend Later Yan, and they successfully defend Shanhaiguan Pass, and fortresses further west like Tangshan.
The Koguryo forces suffer friction with Emperor Feng Hong. The Koguryo King then conspires with the allied Northern Yan generals and inquires if they would turn against Koguryo or submit to Wei if Koguryo were to attack Feng Hong. They inform him that they would not. So Koguryo carries out their plot - the emperor is killed, Northern Yan annexed and the generals accept Koguryo king.
After annexation of Yan, some ministers propose that Koguryo capital be moved to the Later Yan capital in Jinzhou.
The king refuses. From 432, it was clearly shown that the Wei armies could march around the fortresses and mountain passes and attack the capital. Also the capital is much too far from the rich western coast of Korean peninsula. Whereas any Wei or nomad army that wants to attack Yalu valley must also get across Liao river and the hills of Liaodong, so it is too far out of their striking range.
The king is reminded of the plans to move the capital to Pyongyang.
Again he refuses. After the annexation of Northern Yan, those plans are outdated by far. Pyongyang would be too far to be supplied from the wealthy Liaoxi area, or to hear of Wei attacks and react to them.
Finally, someone offers to move the capital to Yalu river mouth. This way, the capital is still separated by the same Liao river and Liaodong hills from any overland nomad or Northern Wei attacks, but tax grain from either Northern Yan areas or west coast of Korea can be brought near the capital by ship.
By 440, Koguryo has capital at the mouth of Yalu, and has repelled a furious attack from Northern Wei. Koguryo still holds Shanhaiguan and some fortresses beyond.
What next?