Individual Investor Order Execution

The quality of execution in the current system varies across brokers because wholesalers offer different discounts to different brokers. The current system is insufficiently transparent but the SEC rule may not improve average quality even if it reduces the dispersion of quality.
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We can't all have better prices. Who loses? Execution is a tiny issue relative to long term rates of return. Actual knowledge of how SEC rules change equilibrium trading is very thin.

Bidding for orders may improve prices to small investors, but fewer orders may go to dealers offering payment for order flow. Spreads on markets may increase.

Very hard to predict what happens, as there are extensive and intensive margins at play. But it's worth trying. If it doesn't work, it can be reversed or modified.

There is a benefit to having institutional investors participate in the auction for retail investor trades. I expect there would be a benefit for high liquidity stocks but could be worse off for lower liquidity stocks. Not clear how the "average" is calculated.
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More competition for orders will help individual investor order execution.

there are competing view on this even among people who have no financial stake in the answer

Potentially, but it's not clear it is a major concern to begin with in equities (as opposed to options), see links
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The direct effect of the rule should be beneficial to retail investors. However, the obvious question relates to indirect effects. I have not seen sufficient research to make conclusions regarding indirect effects - and the magnitude of such effects.

Not clear they will get better prices when they really need them, like when they demand immediacy, since it may take longer to execute under the new rule.

The proposed changes seem likely to enhance competition for retail orders.

Payment for order flow of individual investors makes the payer effectively a monopolist who only has to provide a lower bid-ask spread than the market spread which reflects more informed traders. Competition for clearing individual trades should leave retail investors better off.

Unclear that moving to trade-by-trade competition for some types of order flow improves market outcomes compared to the current lower frequency competition.
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SEC estimates suggest a very small improvement.

New rules will improve price transparency and the new protections on paying for order flow will also protect small investors. The latency rule in auctions is good as well and backed up by research.

I think the rule is unlikely to have a large effect on the average cost of trading for individual investors. It may slightly improve transparency.

One thing we know as "experts" is how much other "experts" really don't know. Just how SEC rules improve or diminish market function is mostly made up. Many cases we do know SEC rules make things worse.

How big is the problem that the SEC is trying to solve with its 399 page proposed rule book? While I like the idea of involving institutional investors in the auction, the market works well now. I would prefer some experiments before making such a big change (ending PFOF).
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Not clear overall market efficiency will be so impacted.

See previous answer.

There will be winners and losers from this rule, but it is very hard to make predictions regarding overall effects. Also, by changing so many things at once, the SEC loses transparency regarding the efficacy of each individual change.

Not all clear and depends what is meant by operation -- on average? during liquidity events? crashes? time of execution? All unclear IMHO.

This regulatory change may increase or decrease the segregation of the trades of individual and institutional investors, and it is theoretically ambiguous which is better for investor welfare.

Markets are very complicated and segmentation of order flow often has equilibrium effects.

Likely to have a positive but very small effect.

I do not think the rule will affect the overall stock market as much to create tangible more efficiency.