It seems that defeating a single boss in the new area is not necessary to get a stronger character, which could even make it OP.

As with most FromSoftware games, difficulty is one of the most fundamental elements of Elden Ring and its recently released expansion, Shadow of the Erdtree. Messmer, for example, or the boss at the end of the DLC, can be the one to cause a few uncomfortable moments (or hours?) for the player, who may find themselves completely unprepared. However, not everyone arrives at the DLC location with the same level and equipment.

The Japanese developers have been working on this, and the Scadutree Fragment and Revered Spirit Ash items can be used to tweak the balance a bit. These items determine how much damage you take and how much damage you can deal. The high difficulty rating for the Elden Ring DLC lead to Steam seeing a slightly lower rating (mixed), so FromSoftware responded by modifying the items in question and their effects (thus changing everything after the patch). Kyle Bosman, who also created a chart on the subject, talks about it in the video embedded below.

This rework was well received by the public, as Elden Ring: Shadow of the Erdtree received a mostly positive rating on Steam (71%). However, the leveling system is tied to a few collectibles, so you could say that you need to defeat bosses to explore the world, but only in certain situations (Shadow Fortress). A map surfaced on Reddit showing that Scadutree Blessing can be leveled up to level 12 without fighting a single boss. The maximum level is 20. Most of the items are either scattered around the map at altars, can be taken from normal enemies, or are just lying around on the ground.

You can get Scadutree Blessing level 12 without killing any bosses
byu/Galdiuz2 inEldenring

In the comments, you can read that you can go up to level 17, so even if you don’t kill the main bosses, you can still go higher.