My experience is that the third sector sector is more similar to the private sector, being driven by achieving results, than either is to the public sector - which is where I've encountered people with a 'that's not in my job description, leaving on the dot, lots of coffee breaks, jobs for life for minimum effort' mentality. (Have seen good public sector people too of course but, a few years ago anyway, it was clearly hard to get rid of dead wood there and there was a culture of not attempting to do so).
My part of the third sector was a specific field, seen as desirable to work in, that attracts very well qualified, capable, motivated professionals, with good degrees and post-grad qualifications, who had made a values-driven choice to work there for lower wages and 'make a difference' rather than work in the private sector and dance to the client's tune. So full of over-qualified, driven people in very low paid, short term jobs, trying to get their foot in the career ladder. Some quite inspiring and many capable people in more senior positions. Some burnout and people driven out by poor employment conditions.
Reading others' posts, it looks like some of that might be peculiar to my field (or a sub-set of fields represented in the third sector).
So my view of downsides (often, not always) are; poor governance, poor or no staff development, high turnover of staff on short contracts, this contributing to poor organisational memory, poor continuity - success not built on through lack of funding, failure not learnt from almost always, so an amount of running in small circles repeating mistakes, also setting up false expectations in partners / users then letting them down, through bursts of short-term enthusiasm which can't be followed up. A tendency to over-promise and rely on staff dedication to deliver.
Also, perhaps especially in small organisations, part of the poor governance is trustees who are passionate about the subject, in which they have huge amateur skill and dedication and cannot quite comprehend that people working professionally in this field don't necessarily want to volunteer in it at the same time - lots of extra unpaid hours, or that they are professionals with careers, not volunteers doing their hobby on a bursary (with a private income / well paid spouse). I think / hope that attitude is (literally, sadly, as they're amazing people if deployed constructively) dying out.
Lack of funds means performance cannot be rewarded financially of course. But, I found there was a willingness in some organisations to do what they could, so offer flexibility to valued employees (all or some).