Are sins discrete events only, or can they also be of a continuous, on-going nature?

Short answer

Can it be argued that sins are only discrete considering we incur fresh guilt daily? Yes, I think so, although terminology influences the answer. Following how the Bible distinguishes sin as deed and sinner as sin-producer out of weakness (as opposed to willful sins by the "wicked", see GotQuestions article Do Christians Sin?) , and following how major Christian traditions develop Biblical theology of stages of salvation where the propensity of sinning post-conversion is mitigated by the Holy Spirit's grace during sanctification (either in Protestant-style simul iustus et peccator or in Catholic-style sinners commit venial sins in the state of grace), then YES, it makes sense to define "sin" as discrete while the weakness that cause Christians to still continuously sin (Rom 7:21-25) until we are fully conformed to Christ who is fully human but without sin (Heb 4:15, 1 Pet 2:2, 1 John 3:5, 2 Cor 5:21, etc.).

Long answer

Your question contains 3 elements that Christian theologies distinguish:

1. sinful acts/deeds (the discrete events)
2. sinful nature (the propensity to do fresh sinful acts), and
3. living in the spirit (the new way of living possible when we are joined to Christ, at first imperfectly, but striving toward the ideal where we no longer do sinful acts/deeds).

The central message of Christianity is that Jesus provides a solution for our souls to be healed so we can die to our sinful nature (#2) and instead have the power so we can NOW live in love (#3), something that we can certainly do perfectly after death. Although different Christian theologies offer different variations of the solution:

• some say perfect / ideal #3 is possible on earth
• some say perfect / ideal #3 is possible only after resurrection of the body
• some say God will give special graces to get much further toward the ideal #3 if we ask, and the evidence is in the lives of the saints / martyrs

they will all point to what St. Paul teaches in the book of Romans, especially in Rom 5:18-Rom 8:17 when he zones in on his observation of how the "power of sin" (#2) works and how the solution (#3) involves:

1. dying to our sinful nature with Christ, cf Rom 6:6

   This is a choice to let God crucify our sinful nature with Christ resulting to our new status in Christ of being a friend of God (Rom 5:11) and a child of God (Rom 8:14). This new status needs to be reaffirmed after discrete sinful acts (#1) in repentance (i.e. asking forgiveness and receiving absolution).

2. making the choice to reject sinful nature (#2) but instead live under the freedom of God's grace and in the spirit (#3) resulting in good deeds for God, cf Rom 6:14, 7:6, 7:4

   This is living consistent with our new status as God's children, possible because we have been resurrected with Christ, but still a struggle given the latent possibility of falling back into sinful nature (#2)

3. persevere in our struggle to keep saying "no" to our sinful nature until we die, the point where we receive the reward: eternal life

   Our daily task is to join forces with the Holy Spirit (Rom 8:2) to be freed from "this life that is dominated by sin and death" (Rom 7:24) so at the end we will receive eternal life, the FREE gift of God for those who chose to die to our sinful nature, given through Christ Jesus our Lord (Rom 6:23).

Conclusion

You ask:

is it a sin to not exhibit a Christian virtue (humility, generosity, etc.) to its fullest? Since none of us mere mortals ever do that, we are therefore sinning the entire time we are in the less than ideal state. We are therefore indeed accruing sin guilt every second of every day.

We have to distinguish between discrete sinful acts and our status before God (justification). Rom 8:1 is clear that once we have accepted Jesus as our Lord and Savior we belong to Christ (becoming slave of God, cf Rom 6:22) and that:

So now there is no condemnation for those who belong to Christ Jesus.

But we need to be careful to REMAIN in Jesus (John 15:4) to produce good deeds for God, for it is possible to fall back to our sinful nature. As long as we retain our status as a child of God, it is OK to commit fresh sinful acts AS LONG AS we don't stay complacent and repent right away when we are conscious of them. BE VERY MINDFUL of the warning given in Rom 6:19 "... which led ever deeper into sin...") so that instead of complacency we should be EXTREMELY conscious of the STRUGGLE toward righteous living through which the Holy Spirit slowly transforms us to the image of Christ (the ideal). I believe God will be pleased if we struggle together with the Holy Spirit in continuously doing a "whack-a-mole" every time a temptation surfaces its ugly head in our consciousness, characterized by St. Paul as "another power within me that is at war with [our minds] ... [ready to make us] a slave to the sin that is still within [us]." (Rom 7:23)