Like most advice surrounding passive investing, this advice may apply in certain cases but the answer really lies in due diligence on the sponsor rather than a broad-brush rule.
As a longtime syndicate sponsor I've been funded by both HNW equity and institutional LPs, so I've seen both sides of this first hand. Most of the deals I've done, and likely the ones I'll will do in the future, have been funded by HNW investors, by choice.
Institutional LPs aren't always what they are cracked-up to be. @Evan Polaski outlined a good case for this above and I agree with what he wrote.
I've seen cases where the LP backed out of the deal at the last minute, not only leaving the operator at the closing table without the capital, but causing the sponsor to lose their earnest money deposit. I had this happen to me once, but fortunately the LP backed out early enough for us to pivot and raise all the equity from our HNW clients. We sold that deal a couple years later for nearly double what we paid, and the LP that backed out later admitted to me that they regret not doing that deal.
An institutional LP can also exercise their control rights to force a sale at a time that may be inopportune for the operator's equity. It wouldn't feel good to lose your equity because the 800-lb gorilla that you are partnered with wishes to rebalance their portfolio and you are powerless to persuade them otherwise.
One time I had an institutional LP in a deal and they were a total pain in the you-know-what. Despite blowing through all performance projections it was just an endless stream of ridiculous requests for reports and information (beyond what was agreed to in the operating agreement) only to satisfy a junior analyst that probably hadn't stepped foot on a property in his very young life. Thankfully I had negotiated a pre-determined buy-out right where I could buy them out at a fixed IRR at any time. I raised additional money from our HNW investors and bought the LP out per the terms of the agreement. This turned out to be a great decision for us and our investors, because a few months after the buyout was complete we sold the property for a massive gain, and our investors received profits that would otherwise have been allocated to the institutional LP had they still been in the deal.
For these reasons but not only these reasons we choose to raise capital from our base of HNW investors. Yes, it's more work up front, but we have an investor relations team dedicated to performing that work so it doesn't distract us from operations. We do get a better fee/split structure, and that's a fair trade for the extra work of managing a large investor base and it's associated extra accounting and tax reporting overhead.
I also find the advice in the OP to be contradictory. On one hand, you advise investors to avoid a GP that "doesn't pass the sniff test", and then on the other hand, you advocate that "there's nothing wrong with investing...with the syndicator who is chasing your checks because you are the only option..." Do you really want to invest with an operator that is un-investable to professional LPs?