Once again, see Freeza on Namek. Characters fighting at a fraction of their actual strength is not an accurate gauge of their stamina; what matters is how long they can fight at full power.
Freeza could fight for extended periods with no problem as long as he was kidding around, but started rapidly losing strength in around three minutes when at full power, then the same thing happened when he was revived, then he worked on his ki control while in Hell to fix it by the time he was brought back for the ToP.
Broly meanwhile never had any need to train and had no one to suggest the idea to him, so it's no wonder that he was running on empty after just a few minutes of fighting at full power.
No, he had increasing strength as he tapped more into his naturally high strength in response to his opponents. Then he capped out in his base form against SSG, capped out his anger mode against SSB, and capped out his SSJ against Gogeta.
This is different from OG Broly, whose energy in LSSJ was naturally filling up to the point it was painful to him and had to be vented out.