Unable to Open Online Account via Safari on Mac

I am able to log-in to the site if I open a "New Private Window."

That is relevant.

Excerpted from If Safari doesn't open a page or work as expected on your Mac - Apple Support,

If that works, use the following steps to remove the website's data, including its caches and cookies. The website can then create new data as needed. If it's a website that you sign in to, make sure that you know your sign-in information before continuing.

1. Choose Safari > Settings (or Preferences), then click Privacy.
2. Click Manage Website Data.
3. Select the affected website from the list shown.
4. Click Remove.
5. Click Done.
6. Open the website again in a non-private browser window.

Try that.

... it must be something peculiar to this particular website, and differences between Safari settings on iPhone vs. Mac.

It probably is. Mac-specific causes often involve proxy servers, VPNs, oddball DNS settings, non-Apple "firewalls" and "security" products including "anti-virus" junk and / or "cleaning" scams... all of them are popular causes people tend to overlook, so be sure to rule out all of them. iPhones and iPads are not typically affected by most of them, which may be the reason their website works on your iPhone.

After doing that, there are a few more troubleshooting steps to follow before concluding the problem is something only the bank can resolve, most likely by changing its page code to be standards-compliant, which you can do nothing about other than to take your business elsewhere (case in point).

First try restarting your Mac in "Safe Mode": How to use safe mode on your Mac - Apple Support. Determine if you can load that page while in that mode. If you can, it suggests something you may be able to fix.

Lastly try the following in an effort to confirm your Mac's hardware is working properly:

1. Shut down your Mac in the usual manner.
2. Load macOS Recovery by holding ⌘ and r (two fingers) while you start your Mac with a third finger.
3. At the macOS Utilities screen, release those two fingers.
4. Choose Get Help Online (or Safari, whichever option appears).
5. Safari will launch, but it will lack your bookmarks, favorites, history and other preferences.

Determine if the problem persists while using Safari in that mode. After that, restart your Mac normally, and reply with your observations.

If it works in Recovery mode, try one more suggestion in an effort to determine whether the problem is something you can fix on your own.

First read How to use the Add Text Feature When Posting Large Amounts of Text, i.e. an Etrecheck Report - Apple Community, in which Old Toad describes downloading and running EtreCheck, a program that gathers various information about your Mac. It summarizes that information in report form, which can then be used in a reply. Any personal information will be redacted. The report may or may not reveal anything useful, but it would be the last step in performing any simple troubleshooting.

More in-depth troubleshooting would require inspecting that website's page code using Safari's normally hidden Develop menu: Use the developer tools in the Develop menu in Safari on Mac. That gets tedious. Even if you were to identify exactly what's wrong, only the bank can fix it anyway, and you're back to step 1.

You can keep the EtreCheck app or just drag it to the Trash when you are finished with it. It does not affect your Mac in any way and does exactly nothing after you quit the app.

Antivirus software: Apple and Bitdefender

😫

I thought I addressed that already.

Like all non-Apple "anti-virus" products "Bitdefender" is junk. Rule 1 of Macs is don't install junk. Get rid of it, all of it.

Follow its uninstallation instructions (assuming they actually work) and run EtreCheck again. Confirm all remnants of it are permanently and irrevocably gone. Never install it, or anything like it, ever again.

Make that line look exactly like the following:

Antivirus software: Apple