If you want to adhere to that, your system is unduly complicated.  

And look, this is EXACTLY what something like the LN is trying to achieve.  You don't have to know what happened in a channel, from the outside.  You don't want to know whether they screwed one-another or not.  You only see the final balance on the block chain.  You don't care what happened in those channels, and whether they did it according to the rules.   What matters is the end balance.

I could even say: this is already the case in bitcoin.  Nobody records the mempool for you if you aren't online.  Miners are supposed to be online.  Yes, PoW allows you to do "offline mining", like in my example, where you do offline mining of a whole year worth of block chain, just to overthrow all that has been done online during a year.  But that's a bad feature of PoW.  

The whole desire to have a system that has "its own offline clock" that can be checked, and to allow someone that has been absent for a whole year, to re-vote everything, is what makes the crypto consensus mechanism unduly heavy, and even prone to attacks that have no reason to exist if you take them to real time online.

After all, the only thing consensus is about, is an agreed-upon decision of what transactions are to be considered valid at a certain point in time, and to come to consensus that all competing transactions after that point in time, will be considered double spends.  That's something that can be judged "on the moment" by those online.  That's way easier.  The only thing that needs to be taken into account, is that due to network delays, if ever there are competing double spends, which one is the one we pick to be the true one.  After a few minutes, we can clearly declare that we've seen all sensible candidates, and pick one in a way everyone will agree upon.  Those that weren't there, simply have to accept that decision.

What screws up most consensus mechanisms, is that there's a reward for proposing consensus.  That complicates matters, because you can develop strategies to obtain the reward.  But if there's no reward, and there shouldn't be any, it is an almost trivial matter if you can take the decision within an online network.

We only have to "confirm the mempool" from time to time.  If you don't get a reward for that, you're not going to compete to do so.  You will do so if others don't, because you have stakes in the good functioning of the system.  If someone else stakes the mem pool slightly differently from you, that doesn't matter much, you can just as well accept his consensus as yours ; it is only a matter of a "symmetry-breaking agreed-upon rule" to decide between you and that other guy, which one is the one to be preferred.