Real-time video encoding is computationally demanding
All of my Windows 7 systems run Windows 10 just fine (including over 10 yr old systems). And most any PC powerful enough for OBS, should be fine for Win10 (the OS can be upgraded legitimately for free, presuming WIn7 licenses isn't known pirated license key). An OS upgrade isn't zero effort, but usually strongly advisable.
Unless you have access to Microsoft's extended premium support (really frickin expensive, some large companies and government agencies may have this, most everyone else.. no), then Win7 has a number of unpatched security vulnerabilities that make Win7 Internet-connected systems really risky.