So, I don't particularly like the rules for Bloodlines in the Birthright Campaign setting; bloodlines are the character-level hook of the setting, and but the default rules don't result in a lot of bloodline powers being acquired.
Now, another book I enjoy is Oriental Adventures. It is incredibly problematic, from the name on down, but there's a lot of fantastic mechanics in there. I love the Hengeyokai (which I cribbed to make the Shapechangers for Hackmaster), and I generalized the Martial Arts rules to apply to the rest of 2e AD&D (and have some ideas about applying to other systems, but it needs more work). And, lastly, the family system from Oriental Adventures neatly dovetails with Birthright's bloodline system, replacing it with the notional "Honor" system of Oriental Adventures.
In Oriental Adventures, a character rolls their family, with adjustments based on class; this system is just a 3d10 roll to determine, which the DM may adjust up or down depending on their campaign's needs (I might suggest a bonus to the roll if you are a Regent, for example, or have holdings). Using the table in the link above (and here), you get a base strength, from 3-30. That base strength also determines how many times and what die you roll on the Ancestry tab, and on the Blood Abilities tab; for example, a roll of 16 on the table means you start with a base blood strength of 16, roll five times with a d100 on the Ancestry table, and 4 times on the Blood Abilities table with a d100 (unless their blood strength increases due to Ancestry).
Unlike Oriental Adventures, all modifiers to blood strength on the Ancestry table are positive. If your ancestor was an Infamous Criminal in Kara-tur, that would be a detriment to your family honor; in Birthright, it means your family DID something, and that usually means that you had more blood strength than usual. Our example character, with the 16, gets 74 (nothing), 70 (Folk Hero, +10), 73 (nothing), 38 (Nothing), and 34 (Ancestral Feud +2)... and my dice get side eye, since I rolled in the 70s three times and 30s twice. Their total Blood strength is now 28. This does not alter their rolls on the Blood ability table, just what their current score is. As the descendant of a Folk Hero, the character receives +2 reaction from commoners when their lineage is known, but receives a -4 on reactions with a specific family (who should probably feature in the game, somewhat).
Now, blood abilities. With a 28, they now have 10 rolls of d100: 84 (nothing), 50 (nothing), 73 (Enhanced Sense, Major), 11 (nothing), 26 (nothing), 06 (roll again +10; result is 64, nothing), 07 (minor power of choice), 02 (nothing), 65 (nothing), 32 (Enhanced Sense, Minor). Despite their advantageous rolls, they wind up with only 3 powers. Since they rolled both Enhanced Sense, Major and Enhanced Sense, Minor, they get a second minor power of choice, from those available to their bloodline.
As they play, they will get additional powers, but it should be noted that they don't get all new rolls every improvement of blood strength; going from 28 to 29 will not give them 10 new rolls, and going from 29 to 30 will only give them 1 additional roll.
I feel this makes for more interesting characters. They have more powers, but their families are more detailed, with histories as distinct or indistinct as the dice determine. Of course, the DM might choose some family events for you, or may allow you to describe your family history without rolls, but these also provide guidelines to do so... a DM might decide that you can have X number of points in Ancestry events, for example, letting players tailor their family histories.