No they shouldn't, for a whole host of reasons.
- How are we going to start flying to Mars without a rocket capable of taking us there?
- These things take a long time to develop. A company that is trying to push the envelope, create markets, and expand human horizons doesn't sit back and wait for others to lead the way.
- SpaceX has shown that they have a pretty good understanding of their market. Certainly better than ours.
- They are developing it with private funding, so it's nobody's business but their own.
I don't know what will happen with Starship, whether it will eventually go to Mars, or be used to haul very large orbital payloads, or what, but I'm sure glad they're building it. You weren't alive during the recent wasted decades with so little forward progress in space exploration. This is a new golden age and we should relish it.