Update LTS to Julia 1.10 - BestieTemplate

## Motivation
Since the recent news that Julia LTS is 1.10, we should update the LTS version to 1.10 instead of 1.6.
This will potentially allow us to get rid of workaround we used to maintain 1.6 and use PackageExtensions instead of Require for optional deps.

This is also a good opportunity to switch the packages to BestieTemplate.

## Tasks

- [ ] Install pre-commit on your machine https://juliabesties.github.io/BestieTemplate.jl/stable/10-full-guide/#pre-commit
- [ ] Add `JuliaFormatter` and `BestieTemplate` in your Julia env
- [ ] Apply BestieTemplate
- [ ] Follow this https://github.com/JuliaSmoothOptimizers/Organization/blob/main/JSOBestie/BestieTemplate/apply-update.jl
- [ ] Fix the conflicts
Remove examples/codes from the README, keep minimal.
Keep README.md and docs/index.md minimal to ease future update.
Rename files in the docs/src folder (must start with a number)
Rename files in the tests (must start with `test-`)
Clean runtests.jl
- [ ] Add extra references in the CITATION.cff (preferred-citation)
- [ ] Run `pre-commit run -a` in the terminal before pushing your branch with the changes
- [ ] Check if some code is specific Julia 1.6 (tests mainly)
- [ ] Create issues to replace Require.jl and use PkgExtensions.
- [ ] Write on Zulip if the doc is not running (probably FORMATER_KEY issue)
- [ ] Write on Zulip if the code coverage is not running (probably FORMATER_KEY issue - or wrong branch tracked)
- [ ] Write on Zulip if allcontributors is not working.
See, https://github.com/tmigot/NonSmoothDynamics.jl/pull/10, for an example.

## Questions (for later)

- [ ] Ask Abel why not use lts instead of mentioning 1.10.
- [ ] How to update? How frequently?
- [ ] Keep JuliaFormatter ? Create an issue on BestieTemplate. Is there a way to do it automatically?
- [ ] Ask Abel if there is a way to keep our modifications in files (example: README.md or docs/index.md)

## Next steps

- [ ] @dpo LinearOperators, RegularizedProblems
- [ ] @MaxenceGollier RegularizedOptimization
- [ ] @tmigot optimization solvers, ADNLPModels
- [ ] Ask Alexis (Krylov.jl, DiffKrylov.jl, KrylovPreconditionner.jl, CUTEst.jl, HSL.jl, NLPModelsJuMP.jl) -> Tangi

Next release is 0.X as updating to Julia 1.10.

## Result
Make an announcement on the website.

## Comments
https://discourse.julialang.org/t/julia-v1-10-7-has-been-released/123141