While the MacBook Pro 13-inch (mid 2010) model could still be useful
and run as new as macOS High Sierra 10.13.6.. You may find it could be
erased via data cable when attached to later model with similar cable.
There also may still be a Recovery partition on the mac drive; have you
tried starting up in macOS Recovery? And then use that version's mac
OS Utilites, to erase the drive? That'd by-pass the issue; as Recovery sees
that internal drive as a separate entity. It can erase, without DVD media.
• How to reinstall macOS from macOS Recovery - Apple Support
https://support.apple.com/en-us/HT204904
Start older MacBook up in TargetDisk Mode.. It then could be seen as
an external drive; to use another mac to erase contents. ~ Then you'd
have a working Disk Utility, the tool to erase drive.
• How to erase a Mac that seems to lack erasure options - macworld
https://www.macworld.com/article/3094986/how-to-erase-a-mac-that-seems-to-lack-erasure-options.html
However with Snow Leopard 10.6.8 installed and able to access Mac
App store online, other later system installers can be downloaded.
(Those do not need optical media. To install a new El Capitan & exit
that before personalizing, would enable a new owner to set it up.
As long old Mac is running Snow Leopard 10.6.8 and sees upgrades.)
To erase the macbook w/o install-restore discs, use Target Disk Mode
and have another Mac's disk utility wipe the drive. Or install a later OS
then it'd be worth more than recycle/trash. You could also remove the
HDD for later disposal or reformatting, etc.
Well, I've gone on awhile here; had stopped to answer phone call,
and other things. There's reason to have older working macbook.
Just got back from half-hour call ~ this sat almost ready to post!
Take care & happy trails!🌞🍀