Some people can. It's not clear that all the people necessary to maintain the current open source ecosystem can. Of course, it's not clear they can make a living by donations either, true! Either way we're talking about something that changes the way resources are currently allocated to shift resources from those with budgets to those maintaining open source.
But, anyway, since anyone can use open source for free, what are open source maintainers going to be invoicing and getting paid for? It seems to me it's not going to be maintaining the software itself -- since you can use the maintained software without paying, anything you pay for maintenance is effectively just a voluntary donation, not a straightforward invoice for work.
So whatever you're invoicing for and getting people to pay is... something other than maintaining the open source software. It seems to me that is not solving the problem of "how do you get maintainers paid", in fact it's taking the people who are well-placed to be maintainers and reallocating their time somewhere else to something they can get paid for, and hoping there's enough net profit they can keep maintaining in their unbilled hours just because they feel like it.
The OP's perspective is more: "How do we get maintainers paid for maintaining? If you are a company with budget who wants to, here's how. If you instead pay someone for doing something other than maintaining, which is often what people end up doing (paying for features or what have you)... that does not actually make maintaining pay, it doesn't work to make maintaining pay."
The usual answer is that customers should pay for "support" that they will not use (effectively a form of insurance), leaving the maintainer free to continue maintaining.
In practice I suspect we'll see a lot more license changes and hostage source in the future.
Getting people to pay for support they won't use sounds to me closer to a voluntary regular donation than it does the "long tradition of setting up a business, having invoices, and providing services for the pay."
I guess it's a way of making a voluntary regular donation seem like a traditional invoice to satisfy the bookkeepers, but that's not what I thought @mfer was getting at. And if it magically worked to produce what are effectively donations, we probably wouldn't be having this conversation.
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But yes, you are talking about something other than open source as the more likely future route, which is why i asked @mfer if they were! That's the straightforward way to get people to pay for software -- make it not open source, so they have to pay to use it.
Continued maintenance of the software is also included under "support". As are custom feature requests, and sometimes marketing/sponsorship (i.e. the company that pays for "support" can request to be mentioned on the website and in the project documentation). All in all, it works quite well.
