EF Core 10 cold start after idle — is using a hosted service for keep-alive acceptable?

You don't need to have a keep alive service. You can set the minimum open connections in your pool via the connection string which should help. below is an example;
Min Pool Size=5;Max Pool Size=200;Connect Timeout=60;Pooling=true;Connection Lifetime=300;

To mitigate cold start performance you it's generally recommended to warmup your database in the startup/program.cs. I've seen various recommendations; either running a very simple select query on your context or simply ensuring the context.Database.CanConnect()