Not bad idea, but won't work because these are designed with analog in mind. When specific voltage hits the phone ringer flashes. You got it right + and - because it is set to direct current not Alternative current.
This is not advisable because I believe when the voltage of around 90VDC hits, it flashes, and that is not safe for doorbell button.
I had mine very customized, and is NOT cheap. It is hard wired and built in my home, you won't see any independent lamp for doorbell. It is hooked up to wall switch, and works even when the light is in off position and yes it does flash. I hate strobe flasher!
Yup, to get phone ringer to flash on doorbell is to use 96v (typical of telephone line "ringing" voltage). IT would be too risky supposed IF you rigged it using 120v serie loop to doorbell button. It'll work just fine until when it's rainy out or High humidy out. IT'll kill anyone pressing the doorbell. You'll get sued by family of zapped loved one. It also mean that They are NOT welcome to your place.
The ONLY way is to take apart phone ringer and modify circuity and add "step down" transformer to 24v DC (using rectifier to power the relay inside of it) plus capacitor and resistors and then serie connect wire to doorbell button then back to hacked phone ringer. IF you know how to do electronics, that's great. IF NOT then find electronics nerd to do it for ya.
IF you have kids or little brother stopped using radio controlled toys, YOu can tear it apart and take transmitter and receiver parts and modify it for wireless doorbell ringer. LOL.
Last Options. SUGGESTION

IF you have TAP (Telephone Assistance Program) that give you $500 voucher in your State, Any vendors that are listed in TAP booklet, YOU can order Sonic Alert DS80 Doorbell/telephone transmitter (one do NOT transmit signal to receiver) $54 OR if you have Sonic Alert receiver, get DS800 Doorbell/Telephone transmitter $84. TAP can only cover $55 and
you can pay the difference. Vendor will get stuck cuz they CANNOT deny you because it's a telephone ringer built with Doorbell ringer. And get urself a new TTY or whatever. (IF you get the voucher request form. Check mark TTY or telephone u wanted and also an signalling device, Be sure you check marked "Telephone strobe" or whatever telephone signalling device. So you can get $500 voucher for any tty under $500 and Telephone ringer for $55 voucher.
That's what I did to replace my 25 years old SA Doorbell/Telephone ringer (I rebuilt it 3rd time and it got worn out badly). So I got brand new through TAP and paid the difference. I love the DS800 Doorbell/Telephone ringer so much and it's so reliable now. And got myself a nice Compact C traveling tty that I can use cell phone or cordless phone to be outside to chat w/ friend on tty LOL.
Good luck
CAtty