On the cycle and scooter to and from Tesco's (using Ferndale road (Bon marche route). But I work in the City so used to use Atlantic Road, Villa Road to get onto Brixton Road mainly.
My Wife is disabled and I need to often use the car to get her around so all of the road closures have added 30 mins to even small journeys to the Pavillion practice. If you are a Brixtionan you will be familiar with the Gresham Road bottleneck and this has a box junction (again often blocked) or the exit from the O2 Academy. Apologies but there are so many.
For the TFL trips, I use Brixton tube to Kings cross which is often closed due to people congestion then walk from there to Angel or I get the bus to Oval to get on the Northern line to city Angel. If I ride to work I tend to ride through the estates then Kennington Park to Kennington road, through the Elephant then up through Borough, spitifields to Barbican then back streets to Angel as there isn't direct bus routes. So it's often scooter to and from work.
If I've understood correctly, as far as cycling is concerned, your problem is mainly with Ferndale Rd and Brixton Rd.
Ferndale is supposed to be one of Lambeth's "Healthy Routes" so if it's now got more motor traffic than it used to, then the scheme is not working. However, it was pointed out a few posts back that the Ferndale scheme had not yet been fully implemented - I don't know if that's still the case. Also, it's normal for there to be additional disruption for a period after things change because it takes a while for people to change their regular journeys and for route planners to update. So would you be willing to give it a few more weeks to settle down, before concluding that conditions on Ferndale Rd have worsened?
As for Brixton Rd I'd agree it's not pleasant to cycle on in the bit just north of Brixton centre. But is your proposed solution here to re-widen the road, to re-instate more lanes? Is your hope that doing this would ease the congestion and traffic levels (which presumably would also help your car journeys which are delayed by having to get through on this road too)? Because generally that doesn't work - if you increase the road capacity it just fills up with more traffic.
If there were some way of getting a proper segregated cycle lane on the N-S axis through Brixton centre then I'd agree that would be very helpful. At the moment, the closest quiet(ish) N-S route is the one that runs down Loughborough Rd and it sounds like you use something like that to get to/from the City, but of course if you are starting from the Ferndale area then you need to get across to that, and it sounds like that's when you use Villa Rd or Atlantic Rd, either of which force you along Brixton Rd. As I understand it, the "Healthy Route" proposed as part of what they call the Loughborough Neighbourhood, which makes its way up Station Rd, and then Barrington Rd is designed to address this and work in conjunction with the Ferndale Rd route. There is still a short section along Brixton Rd that links those two pieces, and I'm not sure if there are plans to make that ink section work better than it does at the moment.
To make a more general point: your example of cycling from Brixton to the city, I think illustrates that there is a quiet cycling route many parts of which are already OK, but that there are problem sections. Even if only 5 or 10% of the route consists of problem sections, that's enough to put people off (understandably). The problem sections are known about, and attempts are being made to address them. But it's a slow and difficult process connecting all these bits up, because you have to make lots of local-level changes to enable it, and every single one of these is going to raise local level objections from people who may not see or value the larger-scale benefits of making these kinds of changes in general.
The same applies to the principle of reducing cut-through routes and reducing road capacity - it seems to work well at a large scale level, reducing the overall level of traffic, but you have to manage to persuade people neighbourhood by neighbourhood that it's worth it, again because the benefits aren't always very visible if you look at it only on a very local level.