I've looked at this stuff a bit in the past couple of years, so here's my echinda. I don't feel like it's easy to cover the information that you need in just a few sentences, so hopefully you don't mind me taking the liberty of writing more than that.
The first thing I'd prioritize is deciding what your goals are in an ethical/ESG ETF (henceforth just "ESG ETF") vs an equivalent "normal" ETF, and also just in general (e.g., investing in an ESG ETF vs no action). If you don't know what your goals are, you won't be able to achieve them. I would assume your goals here are some mix of financial and "ethical", otherwise you'd presumably ask about either ETFs generally (instead of ESG ETFs) or charities.
If your goal is financial out-performance, I'd personally consider it unlikely (all else being equal), though most of the people selling the ESG ETFs will suggest otherwise -- which is, of course, in their financial interest to do. The metric of success that listed companies operate under can essentially be simplified to "maximum profit", so any ESG-aligned actions that also maximize profit would already be incentivized, and anything beyond that will probably incur a performance penalty (unfortunately). Thus, if you accept that premise, logically an ESG ETF is unlikely to outperform in the long term when controlling for other factors.
Some out-performance may occur at times due to the nature of how an ESG ETF will underweight / overweight certain companies and sectors etc. As you've sort of alluded to, there tends to be somewhat of a "tech stocks" bias in some broad-market ESG ETFs I've seen, and since big tech stocks have been doing well, any ETF (ESG or otherwise) that has these stocks overweighted compared to a pure market-cap weighting will have also received that temporary out-performance benefit. The criteria that ESG ETFs use to screen out company exclusions tends to not hit on what big tech does, and tend to be harsher on e.g., the resources sector (mining etc) though it will come down the specifics of the screening process, which differ between different ETF offerings.
If your goal is to "make a difference" by sending more money in the direction of higher-ESG companies, I'd personally consider the effect very minimal. Even if you believe that raising the stock price of higher-ESG companies and/or lowering the stock price of lower-ESG companies improves the company's position in a tangible way (which there is some disagreement on), you're forgetting that you're not the only actor in the stock market. To quote my own comment from a few weeks ago:
The problem with doing this from an "activist" approach is that you're doing it in a market where the primary incentive of most of the other actors is monetary. They basically just want whatever makes the most money.
Let's say that the entire stock market is 3 companies: Ace, Brilliant, and Catastrophe, and the share price of each is $10. If a bloc of "green investors" switches their portfolios to exclude Catastrophe due to the company's environmental impact, the share price of Ace and Brilliant goes up, and the share price of catastrophe goes down. However, the majority of this effect is wiped out by other investors, who now see that Ace and Brilliant are overvalued on a purely monetary level, and Catastrophe is cheap - so the market corrects back to either the status quo (where all three are $10), or something very close to it.
You may be able to have some non-zero impact by virtue of changing how your shareholder vote is used, if your ESG ETF provider votes "better" on their ESG ETF offerings (seems unlikely), or just in general votes "better" than providers without ESG ETFs. In my experience, evaluating this is hellishly difficult as an individual retail investor because you have to sift through thousands upon thousands of shareholder votes, and I'm not personally aware of resource that solves this problem for individual investors (though that doesn't necessarily mean that one doesn't exist somewhere!). Given the uncertainty, and the little information I do have about shareholder voting done via the ESG ETF providers that I'm aware of, I think there is likely very little effect here, on average, though any provider that explicitly notes that they vote "better" is presumably more likely to have a non-zero effect on the outcome of shareholder votes if more people buy the relevant ETF(s).
(Continued in replies since there's a 5k character limit and I guess this is an extended infodump.)