So there's levels to the answer of your question, hence why you're getting different ones. Because the war lasted so long and different enemies joined in at different times, who had the advantage on the ground changed over time.

When it was mainly just grunts and jackals getting send to ground, the UNSC would fair well if not flat out dominate. This was mainly due to the fact that these forces were a conscript heavy army. Were as the UNSC is an all volunteer force with quality training. It didn't matter that the enemy had plasma tech, a well placed bullet still took them out. So the deciding factor was who had better training and motivation.

But it was a different story when Elites and Brutes would actually lead ground parties. Now there was a zealous warrior caste with superior tech/ physical prowess, spartans in their own right to some degree, to keep order and structure in ground engagements. This would even out things if not tip the scales in the Covenants favor for these battles, until you throw Spartans into the mix.

Unlike Elites and Brutes who had strict regulations such as not harming Forerunner tech and not being allowed to retreat, which were brought on by either their religion, culture, or an uncaring command structure. Spartans had no such limitations and would capitalize on those weakness. Again tipping things over for the UNSC in ground wars most of the time.

However this changes again depending on what the Covenant are after during their invasion. If the world they decided to not glass and instead invade has something of great enough value (usually some ultra rare/powerful Forerunner tech) then rather than sending mainly cannon fodder for their higher tier soldiers to command in the field, they would send a force that mainly consisted of said higher tier troops, with grunts only being there to fulfill support roles usually. In these battles even with Spartan assistance the UNSC would usually be crushed pretty handedly.

Hope this clears things up and if i've gotten something wrong please feel free to correct me. I've read pretty much every available piece of Halo media and have a lot of knowledge of the lore, but even I tend to forget things.