If you really interrogate what you are trying to get out of your formal assessments, I wonder what you'd find.
The true purpose of a formal assessment is to find out what pupils can do, and to show the progress that has been made over a period of time. The true purpose is summative assessment.
Yet, when we dig down, we might also be expecting more of those formal assessments. For example, we might be wanting them to:
Act as a formative assessment, for example, identifying strengths and weaknesses to inform future planning and teaching.
Provide teachers with a tool to support their own ongoing assessment, particularly for teachers who are less confident in their own assessment.
Provide consistency across a team of teachers - this is often the desire of a leader who might want to insure against discrepancies between different teachers' assessment outcomes.
Act as a form of test or exam preparation for pupils following the logic that if they're doing a test, then it might as well represent the layout and requirements of SATs or GCSEs.
And, depending on the school, there could be many more additional puposes for formal assessments.
The thing is, if a formal assessment begins to stray into any of the above it could affect several things, such as:
Performance - if it is seen to be high stakes, for example, then pupils may be more worried as the complete it, this potentially affecting their outcome and giving an incorrect picture of what they can do and what they know.
Integrity - if the assessment is designed to do anything other than ascertain progress and attainment, there is the risk that it doesn't fulfil its core purpose as well. For example, if part of its design is to be a support to teachers it could conceivably become less about assessment.
Teacher assessment - if the assessments are designed to act as a support to teachers and to ensure consistency, this could be to the detriment of other forms of assessment. School leaders would need to make sure that staff were trained accordingly so as not to only rely on the formal assessments: more information, and more nuanced information, can be gained over time in class than can be seen in a single-point-in-time formal assessment.
It may be better, if you can, to reduce the number of purposes your formal assessments have so as to focus in on the main thing: finding out what pupils have previously learned.
In order to pick up some of the other tasks you've removed, there are other things you can do, for example:
Provide teachers with materials, training and support on how to assess learning in an ongoing manner.
Incorporate familiarisation with what tests look like into everyday lessons.
Use shorter, more regular formative assessment techniques.