For example, if one had to pay $1000 to the medical service provider for a procedure, and upon making a claim - only $300 got covered by private health insurance, does one preserve both receipts for audit purposes, and simply claim additional $700 (assuming there are other expenses) on their total health related tax deduction in Canada?
On the medical expenses page, https://www.canada.ca/en/revenue-agency/services/forms-publications/publications/rc4065/medical-expenses.html, the CRA says (about half way down that long page)
For all expenses, you can only claim the part of the expense that you have not been and will not be reimbursed for. However, you can claim all of the expense if the reimbursement is included in your income (such as a benefit shown on a T4, Statement of Remuneration Paid, slip) and you did not deduct it anywhere else on your tax return.
So yes, if something cost $1000 but you were reimbursed 300 you claim 700.
The thing about being on your T4 could only happen if your employer reimbursed you, not private insurance.
If it worries you that the quote above is in the middle of a section on travel expenses, there is also this, in a list of what you can't claim:
the part of medical expenses (including travel expenses) for which you can get reimbursed, such as reimbursements from a private insurance
Note it only says "the part" that is reimbursed can't be claimed. The rest can.