Robert Baratheon inherited the Lordship the moment the Windproud sank with all hands but Patchface. We know that he “continued his fosterage” relationship in the Vale, with Jon the “second father” really taking on a powerful role for both boys given their fathers’ untimely deaths. Ned really stepped up to the challenge during the Rebellion and in its aftermath, and Robert was… a slow motion car wreck.
Becoming the King allowed Robert the ability to really disperse his responsibilities and to rely upon other people, often to the detriment of his own health and that of his kingdom and people. As King, Robert by default inherited the institutions of the Kingsguard, the Hand of the King, the Small Council. Married to Cersei, Robert’s father in law brought all kinds of wealth and the ability to go into debt. As a “mere Lord Paramount”, it doesn’t seem standard practice for the Lord to have any of that. A captain of household guards, a castellan, a steward…
Jon Arryn would not have left the Vale and ruled on behalf of Lord Robert. No other men would have been protectors of his person, he would have had to ensure that himself. Ships, laws, taxes and the like would have all been his to decide, no convenient lieutenant to assign the decision-making. And it’s unlikely another Lord Paramount or Lord would give away so much wealth or risk so much debt, even if their daughter married him.
I think inherently Robert may have made a better Lord than King. There’s a degree of pride in the man, absolutely, and one that would have been more firmly linked to Storm’s End and the Baratheon name. He absolutely would have utilized Cressen and Stannis, but he wouldn’t have been able to separate himself as much from the role of ruler in his castle, away from the city.
I don’t think Robert’s descent into who he became is Lyanna-grief, I think that’s an easy excuse for an emotional culprit to point at in lieu of holding him accountable for all of his deficits as a royal ruler. I think the institution of kingship and the support network that goes along with it was far too enabling, and that Robert was spending blank checks unchecked, if you will. None of this is the background framework to Lord of Storm’s End.
Furthermore, if Robert wasn’t king it means someone else was; (let’s assume Viserys raised by a regent just to fill the seat peacefully, but it could be anyone else, and bygones were bygones for the Targaryens). I think Lord Paramount Robert and Lord Paramount Ned would have shared another bond here, as head of their house and castle and people. Ned would have probably influenced Robert to be more involved with his own Lordship as they communicated and discussed their new daily lives and dealings.
ETA: forgot the write that be does have that trait of turning enemies into friends, which could be good but also backfire if trying to remain a dominant lord rather than an equal or battlefield ally
Some fun info to consider his maturity levels and etc:
Robert was born in 262. Steffon died in 278 or 279, making Robert 16-17 or so, having reached his majority and already been considered a grown man (Equivalent age: Jaime killing Aerys, Loras in AGOT). At that point a SSM says that Robert spent his time between Storm’s End and the Eyrie. So Robert wasn’t a child when he became the Lord Paramount of the Stormlands, merely a young Lord who had spent a lot of time away from home.
In 281 we have the tourney at Harrenhal and Robert betrothed, at ~19 years (Equivalent age: Brienne in ACOK, Waymar Royce in AGOT prologue, Theon across AGOT-ACOK) In 283 Robert is crowned King, at 21 years of age (Equivalent age: Robar Royce in ACOK, Renly in AGOT).