I've missed the tax filing deadline, so at this point does it make any sense to file for an extension?

It's too late to file the extension.  Just file as soon as you can.  You can efile until Oct 16.  If you are getting a refund there is no penalty for filing late.

You are too late to request an extension.  You had to do that no later than deadline day--April 18.   If you do not owe or are getting a refund, there is no penalty for missing the deadline to file.   If you owe tax due, you are now subject to late penalties and interest which will be billed by the IRS.  TurboTax will not calculate that. 

E-filing and online preparation will be available until mid-October 2023; after that, online and e-filing for 2022 returns closes down and you have to file using the CD/download an mail the return.