Hi,
First post on this forum and I hope it will help you getting an idea on how to do what you want.
I've had the same exact experience and I've built a flow a month ago to be able to get requests from SHIFTS, approve/deny the requests and send an outlook notification with an event invite attached with the time-off information. It may be not elegant but it works.
Here's an overall picture of the flow:

First I'm using a time trigger to check for new requests every 30 min and filter for only pending Time-off request.

'TimeOff Request state' is equal to 'pending'
Then a few boxes to convert time and prepare for what will be needed for the ICS invite:

Next is a filter to convert the Time-off request reason string to it corresponding text followed by a box to identify the user who has make the request.

'List of time off reasons associated with a team Single time off' is equal to the filtered output of the pending request 'TimeOff Request TimeOff Reason ID'
Now I'm sending the approval request via email to whom will decide. Take note that some basic html will be needed to get the email body formatted correctly. And the condition box will branch-off accordingly to the approval decision.

If the time-off request is approved, it will be added to SHIFTS, otherwise it will be rejected and a notification will be sent to the user.

If the time-off request is approved, an ICS template and file creation will be done in a temporary placeholder on a sharepoint site. Some data manipulation is needed for DTSTART and DTEND to get the time format to ICS standard.

DTSTART and DTEND need to be formatted to ICS, I do it with an Expression:
replace(replace(outputs('Request_startDateTime'),'-',''),':','')
And finally, I'm sending an email notification to the user that his request has been added to SHIFTS and that he can use the ICS invite attached to the email as a reminder in outlook calendar.

I've worked many hours on this as I'm no coder and I hope it may help you or anyone with a similar request.
K.
Don't forget to mark as a solution and thumbs up if you like.