What defines "the best" and what creates true competition is quite fickle. The private equity market, despite all recent changes, is still inherently inefficient. So I'd venture to say there are several dynamics going on:
1) Social Proof: Many investors are just lazy and won't look beyond who is introducing or vouching for the deal. Nothing wrong with that, it is a legitimate defense mechanism, but this limits what they actually pay attention to and creates frenzy around half-baked projects with "superstars" onboard. Think Color.
2) Right Fad: This is closely related to the previous point. If an investor has a theory that something "just like XYZ" is the next big thing, they might be willing to overlook a LOT of legitimate red flags. This could create a frenzy around a certain space or make them avoid another because "it is played out", which is often wrong.
3) Their Knowledge: On one hand this is what they legitimately understand about the space to make their own intelligent judgement of why there is an opportunity. That is the type of investor with the most potential to bring something useful if they can explain: "because I've done this and this, I can contribute this". Flip side, experience often means pre-conceived notions, with Exhibit #1 being YC's bias against solo founders.
4) Actual Metrics: If your numbers speak for themselves quite a few investors would want in just on this basis. The flip side it might be too late. Another flip side is investors might think a deal is overpriced or they cannot get enough control, etc. Best companies though are able to execute regardless of who invests or not invests. That's where you can have a legitimate competition on who is allowed to invest, based on what they offer.
5) Hidden Gems: Investors can often get the best deals by being early enough but that's where you have to find something new, that has a good chance of success, bring something to the table and get a better deal. But this requires to pay attention and do some actual work, so not everyone wants to do this vs. try to co-invest in the latest syndicate someone else is putting together.
If you are an entrepreneur you want to maximize your outcome, but if you focus all your time on jumping through the hoops of investors who do not actually understand what you do or have much to contribute you are being distracted from actual business. Note, I am not using the word "startup" here, because every successful startup is a business. Ignoring this simple fact is what gets people into trouble, leads to bad deals or puts them on the road to failure.
So yes, investors pitching to entrepreneurs could be a great idea but it works only if both parties bring the right mindset to the table and can articulate their value proposition.
PS: As entrepreneur you have to be extra careful with people who invest mainly based on social proof / fads. Do not expect them to do much for you, though their involvement might be a social proof for someone else who would make an actual contribution.