As for Lampard, I'm sure the media (and possibly some of the fanbase) perception will be that this is a rash decision, but this was inevitable. Frank simply did not have the skins on the wall that he can point to and say, we are going to get out of this, I've been through it before and won trophies, or whatever. The only way he was ever going to stay at Chelsea would have been to win the Premier League or Champions League very fast, always improving with no down period like this. Now we have come to a down period, and Frank cannot say he's been through this before and come back stronger. This is not a bad patch in an illustrious career, because Frank is so inexperienced it also looks suspiciously like it might just be the standard mediocrity that you should normally expect out of his level of ability. It was a mistake to give Lampard the job as soon as Chelsea did, and it was a mistake for Lampard to take it, IMO, but I understand why both sides gave it a shot.
The pandemic probably hasn't helped either, I imagine another year without Champions League is probably a pretty dire prospect for a club financially, especially without having the gate revenue right now.