It really depends on the system. In Chinese straight sword, there is a some difference in the range of techniques practiced, which are handed down from teacher to student, from an extant system, and the practice of straight sword for competition sparring in reconstructed systems. This is partly due to the range of techniques that can be applied in a given sport, vs. the real world combat these sports emulate to some degree.
Chinese straight sword transformed into a non-military sport/exercise only in the 20th century, when swords were used less and less in combat, before disappearing entirely from the battlefield.
I don't think lineage would be a major factor in sports like MMA or boxing—only results in the ring.
An archaic martial art like straight sword in the contemporary era is mostly about preserving the tradition and knowledge, even as it no longer has realistic, direct practical application in any likely scenario.
One clear advantage to studying within a validated lineage in this regard is there is generally rigorous training in the basics, because those basics have been reinforced and refined for several generations. Additionally, the teacher may still be receiving instruction and correction from their teacher.
By contrast, a boxer with a good coach is going to get this without a martial lineage, and this also seems the case for judo and MMA.
In the wider martial arts it's more of a crapshoot, such that you see a lot of sloppy practice, which can derive from inadequate instruction. A validated lineage can reduce the chances of this.
- Lineage does not guarantee quality
Martial arts is also a business, and, like any franchise, as the number of schools rise, the quality of any given school can vary. (Ideally, branch schools are connected to parent schools, such that newer teachers can themselves receive correction over decades.)
Much of the martial arts knowledge is passed down directly, or as part of an oral tradition, such that, prior to the 20th century, documentary evidence to validate claims is scant.
This isn't a major problem, as any given martial art must be continuously tested, these days typically in a sport context. But it is a potential pitfall for those starting out in the martial arts, before they have the experience to evaluate what is high quality and reliable, and what is not.
- Lineage can be an important consideration, depending on goals, or not a consideration at all
Lineage seems to be most important in Chinese martial arts, where tracing the history and a given style or movement, as they evolve, is valued.
Chinese martial arts potentially give highest weight to the "art" dimension, where there was never a real separation from performing arts. (Chinese Opera and sword dancing, originally military training exercise, known today as "wushu forms", directly leads to Kung Fu movies, starting in the 1920's. Most contemporary fight choreography derives from this form, and most of the action direction and editing related to complex fight scenes in the visual medium in general, because most of the early technical cinematographic work was done in Hong Kong in more recent decades.)
Even Chinese martial choreography has lineages, most famously the Yuen Clan and the Lau. Lau Kar-leung traces his martial lineage directly to Wong Fei-Hung through his father Lau Cham, student of Lam Sai-wing. Gordon Liu learned Hung Ga from the Lau.
In Tai Chi, Chen style is though to be the oldest, and Yang style the most widely practiced, with at least 6 or 7 other major styles. It's useful to know where an application comes from, and why it may be practiced differently in different styles (most tai chi movements have multiple applications.)
Lineage in the Chinese sense doesn't seem to be a factor in the highest prestige modern martial sports—boxing, judo and MMA—with perhaps a caveat for BJJ, which is the product of multiple lineages over multiple generations, leading to modern MMA.
I have heard the Kodokan spoken of as a gold standard, and Jimmy Pedro and Kayla Harrison talk about the value of this resource and connection for US judo.
BJJ itself arises out of the Kodokan, via Mitsuyo Maeda through Carlos Gracie.
Muhammad Ali and Sugar Ray Leonard would probably say they are "students" of Sugar Ray Robinson, as scholars of the "sweet science", meaning he was potentially their greatest influence, as both on on record that Robinson was the greatest of all time. Both credit Robinson for providing the platform they themselves stood on to continue advancing the art.