[Why did mighty lords of Mace Tyrell and Paxter Redwyne's calibre waste their time and efforts in besieging an untested young lord with (apparently) only a few thousand men (and those weakened more and more of hunger to boot)? Meanwhile their overlord were losing the war?]
The Targaryens had lost a number of battles (and had also won some), but they weren't really losing the war until the Trident and the Sack of King's Landing. And then it was lost. And sieges were a crucial part of medieval warfare. Storm's End was not geographically strategic, but it was the base of Robert's power, as important to House Baratheon as Winterfell was to the Starks. If it had fallen, Robert would have lost his home and his lands... and two of his brothers would have been hostages in enemy hands. All important chips. Also the fall of Storm's End might have convinced many of the storm lords supporting him that the time had come to bend the knee. So the castle was hardly unimportant.
Tyrell had a sizeable host, but some of his strength was with Rhaegar, certainly. Rhaegar actually outnumbered Robert on the Trident, although Robert's troops were more battle-tested. I haven't gone into the whole history of the fighting, but there was a good deal more to it than just two armies meeting on the Trident. There were a number of earlier battles, sieges, escapes, ambushes, duels, and forays, and fighting in places as farflung as the Vale and the Dornish Marches.
[The other possibility is that Mace Tyrell thought it a good idea if Mad King Aerys died, but would not take the chance of actively moving against him. Instead, he stayed put at Storm's End, still appearing for the world to be on Aerys' side, while silently hoping for his death. When Ned appeared, he dipped his banners quickly enough.]
When Ned appeared, Aerys, Rhaegar, and Aegon were dead, and Viserys fled. There was no one left to fight for, and the war was clearly lost anyway.
The modern concept of "total war" really didn't exist in the medieval period. Armies were personal, as were loyalties. The leader who wanted to fight on till the last drop of blood might well have found himself fighting on alone, since his vassals were likely to have better sense, and their levies were more likely to follow their own lord than the "general." Tyrell's surrender was pretty much warfare as usual. If he had =tried= to give battle to Ned in a lost cause, he might well have found his more opportunistic bannermen deserting to the other side.
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Инцест инцестом, а рыцарство это святое.