I imagine your friends who couldn’t be bothered getting annulments are not likely to care about wanting to bother with any other process that can help with their growth and healing of their circumstances either, so no, they aren’t the people that the Church is mostly addressing as pastorally wanting. There is also no suggestion that there should be a change in the general rule for everyone in 2nd marriages.
It is not that they are too lazy to apply. They truly believe that their first marriage was valid. They believe that seeking an annulment would be lying. In their minds, marriages sometimes break down. It’s the concept of “falling out of love”. If you read what Cardinal Kasper and others are proposing, it includes adopting the Orthodox practice of
oikonomia, where by the Church would tolerate second or even third marriages.
The types of situations that the parish priests have found needs deeper examination are those who simply have no recourse to any other option but to end the second marriage if they are to be included in Church life. It is not about condoning a second invalid marriage. It is about recognising the desire to heal and grow as being a ritual part of the penitential process.
So the circumstances you have been beating around the bush about would be someone who remarried and later applied for and was denied an annulment?
If so the Church already allows for an option where the cannot separate. They
must cease the carnal relationship. But that’s too hard, some will complain. Would we tell a married woman with a lover on the side the same thing? What is she was deeply invested? How about the man who leads a double life with his mistress and their children? Adultery is adultery and to treat it otherwise denies the clear teachings of Christ.
The problem is that we are talking about ignoring a pretty major sin for pastoral reasons. This undermines the very core of marriage. Simply feel bad enough and magically every ongoing and future act of adultery will be forgiven.
In essence you either have to change “the rules” to allow certain people to continue in grave sin to still be in full communion or you have to accept that persons can confess and receive absolution for sins they have no intent to amend. Either requires not just pastoral changes, but also include huge theological repercussions.
It leads us down the rabbit hole where we start asking what other grave sins should people be able to persist in yet still be counted in good standing.