Cheapest and simplest will be a phone.
An old Nokia S60 with GPS phone off eBay for a few quid, battery will last ages, just install Endomondo or Sports Tracker and export GPX to import to Strava.
Or cheap Android which will run Strava (battery life maybe not so good).
Frankly a phone GPS chip will be much the same as most cheap GPS devices. Though can understand desire to have a dedicated device. I went Garmin to avoid sucking the battery dry in no time on my phone, which I want to be able to use for emergencies, check where we’ve lost people, etc. Though I did used to carry my old Nokia for GPS while also having my main phone.
Another option is a Bluetooth GPS device. I had one that I think was about £20. No display, it’s just GPS. I used it with phones that had no GPS but it might save battery on the phone if the phone GPS is battery hungry. Then again the Bluetooth will suck battery instead. Needs additional software on the phone to work also.