BUG : pandas.DataFrame.to_excel function not working in pandas 1.3.1
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#### Code Sample, a copy-pastable example

```python
# Your code here
writer = pd.ExcelWriter('existingFile.xlsx', mode='a')
df.to_excel(writer, sheet_name="existingSheet", startrow=writer.sheets["existingSheet"].max_row, index=False, header=False)
```

#### Problem description



It seems this function is broken in pandas 1.3.0, when trying to append to an existing sheet it will either raise an error, create a new one or replace it so it is impossible to append a data frame to an existing sheet. according to this [document](https://pandas.pydata.org/docs/reference/api/pandas.ExcelWriter.html)

> if_sheet_exists{‘error’, ‘new’, ‘replace’}, default ‘error’
> How to behave when trying to write to a sheet that already exists (append mode only).
> 
> error: raise a ValueError.
> 
> new: Create a new sheet, with a name determined by the engine.
> 
> replace: Delete the contents of the sheet before writing to it.
> 
> New in version 1.3.0


#### Expected Output
Able to append to an existing sheet without error
#### Output of ``pd.show_versions()``

<details>

[paste the output of ``pd.show_versions()`` here leaving a blank line after the details tag]

</details>
