Before you can answer that question, you need to decide what your objective is, because you will only know what types of testing to do and which issues to fix when you have done so. It might be one of many things, such as:
- Doing the best you can in the available time with the available skills and no budget for professional support.
- Achieving conformance with a specific WCAG version, usually 2.2 level AA.
- Achieving conformance with some other accessibility specification, such as EN 301 549.
- Achieving the best possible user experience for people with a wide range of disabilities.
- Protecting yourself from the threat of legal action. Realistically, this is only an issue for organisations in the US.
- Doing no more than meeting your contractual obligations, whatever they are.
If the answer is anything other than the first one, you will need a substantial amount of professional support. A lot of testers will “have a go” at accessibility testing, but to do it properly requires a wide range of skills and knowledge and thousands of hours of experience and support from senior accessibility consultants.
Experience
To put that in context, we don’t employ anyone with less than 10 years’ full time accessibility testing experience, which we regard as entry level. People at that level still make a lot of mistakes and their work needs to be checked and fixed by someone much more senior.
The tests you mention are a start, but I don’t like to see people testing with screen readers and other assistive technologies unless they have a lot of experience of running or observing user testing sessions with disabled participants. Without that experience, you have no idea what will and won’t be a problem.
Learning
Self-learning is a poor strategy. We had no choice 20 years ago, but there are some good training courses now. Take a look at Deque University, the IAAP WAS course and Sara Soueidan’s course at https://practical-accessibility.today/
Tools
FWIW, automated tools don’t find anything like 55% of the issues - it’s closer to 25%. You probably got that figure from a bullshit Deque article that claimed 57% based on a completely invalid analysis.
Accurate WCAG conformance testing can only be done by inspection of the code and user interface. You don’t need any assistive technologies to do it. But we do use a wide range of single-purpose bookmarklets to help with it. However, I am reluctant to recommend any because pretty much every one of them has bugs or unexpected behaviours, so you have to be able to recognise when they are giving the wrong result. That equally applies to larger automated testing tools, all of which are buggy.