Hi DanielCossey,
September is the main start date for most PhD students. You can in theory start a PhD anytime of the year but most universities will push you towards 3 or 4 start dates (September, January, June mainly). So if you can wait until September 2022 you will have far more funded opportunities. If you need start to mid 2022, you might find that other PhDs have unofficial flexible start dates. A lot of PhD supervisors can start the PhD a few months earlier or later if you ask for it but it depends on the funding.
Self-written projects have a longer application process that requires the early application date. It is also more competitive. I don't have any experience writing a PhD proposal but I think its difficult unless you have a research area in mind already. Out of curiosity what field are you in?