Looking at flights from SFO to LHR, there are 2 flights in the next 24 hours, both of which have Basic Economy (at a one-way fare of $528, it's great to fly off-season.)

You are unfortunately correct re seat assignment, this says that United will assign the seat for you: https://www.united.com/en/us/fly/tra...c-economy.html

This does vary from airline to airline, I know from experience that both Virgin Atlantic and KLM allow seat selection by the passenger at OLCI time.

UPDATE: As a test I started the booking dialog for the flight that was less than 24 hours away, and then the flight that is more than 24 hours away.

For the flight that was less than 24 hours away, it offered me seats for free. For the flight that was more than 24 hours away (27 hours actually), it wanted to charge me for seats.