Just gonna post my thoughts on all of this:
Do not offer funds to people that placed below 5-8th.
I think this makes sense in a world where players are receiving invites to LoTS. If you were doing 4 quals, it makes sense to reward people for winning the quals.
Adjust group stage so captains choose their groups.
This is good, I think having higher seeds choose their groups first worked well for Spring Invitational. It led to more balanced groups overall and better playoffs later.
Have group stages take place on differing days for more coverage.
I think this is a good idea if you have the casters for it. I think spilting the group stage over 2 days and having the groups fight at different times makes sense from a viewer perspective. It also allows certain casters to cast groups that they aren't playing in on FAFlive. I think you should try to avoid having 4 different streams going at once for each group though.
Maybe include map gen in some way.
I think map gen was pretty successful in the Spring Invitational and having a few map gen options in a map pool to let players opt in to map gen goes a long way.
Option A: 12 invites, 4 qualifiers
This is strongly my preference, there isn't really a need to do invites and reserved slots. The point system should reflect who the top of competitive faf. Those high level players who only want to play one tournament a year should easily be able to secure a top 4 slot in the qualifier.
Point System Currently:
The distribution of points between tournaments seems solid but I would consider increasing the points for 2020 LoTS if you go with no reservations. I would probably double it and give 4-8 points for 9-16th. Especially considering qualifying for 2020 LotS was more difficult than qualifying for 2021 Spring Invitational.
Tournament Date Logistics
I have Summer Invitational currently penciled in for July 10th, July 17th, and July 24-25th and Fall Invitational scribbled in for September 18th, 25th, and October 2nd and 3rd.
Overall I think this is a good direction for LotS to go. I think players sometimes miss the point that the FAF patreon (and the money that people such as myself donate) is being used to facilitate a high level competitive scene, not to just give money to the top players. If players aren't participating in the high level competitive scene, they shouldn't expect the tournaments to cater to them.
This is an important step to reviving the health of the high level tournament scene which is a useful marketing tool for FAF as well as good for broader community building.