Another name for these is the capital sins or vices, meaning that they are the root causes of other sins (from the Latin caput, "head", also meaning "principal"). Gregory the Great wrote in his Commentary on Job (the Magna Moralia) 31:88:
These several sins have each their army against us.
For from vain glory there arise disobedience, boasting, hypocrisy, contentions, obstinacies, discords, and the presumptions of novelties.
From envy there spring hatred, whispering, detraction, exultation at the misfortunes of a neighbour, and affliction at his prosperity.
From anger are produced strifes, swelling of mind, insults, clamour, indignation, blasphemies.
From melancholy there arise malice, rancour, cowardice, despair, slothfulness in fulfilling the commands, and a wandering of the mind on unlawful objects.
From avarice there spring treachery, fraud, deceit, perjury, restlessness, violence, and hardnesses of heart against compassion.
From gluttony are propagated foolish mirth, scurrility, uncleanness, babbling, dulness of sense in understanding.
From lust are generated blindness of mind, inconsiderateness, inconstancy, precipitation, self-love, hatred of God, affection for this present world, but dread or despair of that which is to come.
Because, therefore, seven principal vices produce from themselves so great a multitude of vices, when they reach the heart, they bring, as it were, the bands of an army after them.
Thomas Aquinas agreed (Summa Theologica I-II Q84:3):
A capital vice is one from which other vices arise, chiefly by being their final cause.
He concurred with Gregory's enumeration. He does not link these vices with the venial/mortal scheme; in Q88 of the same part he describes the distinction as "mortal and venial are mutually opposed as reparable and irreparable", although he admits that venial sin can become or lead to mortal sin.
Dorothy L. Sayers, in her introduction to her translation of Dante's Purgatorio, described the vices rather neatly as "the fundamental bad habits of mind recognized and defined by the Church as the well-heads from which all sinful behaviour ultimately springs." She continues, in the manner of Gregory:
In classifying sin under these seven main heads, the Church displays more subtlety and a profounder psychology than is sometimes supposed. It is, for instance, often asked: "Why does the Church not count Cruelty as a Deadly Sin?" The answer is that although cruelty is indeed (in one sense) a sin deadly to the soul that indulges in it, it is not a root-sin. No sane person is cruel for cruelty's sake: there is always, hidden behind the act and habit of cruelty, some other (often unacknowledged and unsuspected) evil motive. [...] It may, in fact, derive from any one of the Capital Sins: from sheer selfish indifference to others' needs and feelings (Pride); from jealousy, resentment, or fear (Envy); from ill-temper, vindictiveness, or violent imagination (Wrath); from laziness, cowardice, lack of imagination, complacency, or irresponsibility (Sloth); from meanness, acquisitiveness, or the determination to get on in life (Avarice); from self-indulgence and the wanton pursuit of pleasure (Gluttony); from perversions of sexual and personal relationships, such as sadism, masochism, or possessiveness (Lust).
In summary, they are quite different from the Ten Commandments, which are about specific categories of action; the seven capital vices are categories of sinful thought which can lead to sinful deeds. As you note, there is no real basis in the Bible for this exact catalogue (there are various sin-lists but not this one). And there's no reason to believe (nor, I think, do the early writers claim) that the list is in any way "complete". I would imagine that it has persisted simply because it has been found pastorally useful, since the seven deadly sins are easy to preach about, and memorable for the congregation.