The question is difficult to answer, because it is so indicative of a general confusion and muddled state-of-affairs in much of the meta-analytic literature (the OP is not to blame here -- it's the literature and the description of the methods, models, and assumptions that is often a mess).
But to make a long story short: No, if you want to combine a bunch of estimates (that quantify some sort of effect, a degree of association, or some other outcome deemed to be relevant) and it is sensible to combine those numbers, then you could just take their (unweighted) average and that would be perfectly fine. Nothing wrong with that and under the models we typically assume when we conduct a meta-analysis, this even gives you an unbiased estimate (assuming that the estimates themselves are unbiased). So, no, you don't need the sampling variances to combine the estimates.
So why is inverse-variance weighting almost synonymous with actually doing a meta-analysis? This has to do with the general idea that we attach more credibility to large studies (with smaller sampling variances) than smaller studies (with larger sampling variances). In fact, under the assumptions of the usual models, using inverse-variance weighting leads to the uniformly minimum variance unbiased estimator (UMVUE) -- well, kind of, again assuming unbiased estimates and ignoring the fact that the sampling variances are actually often not exactly know, but are estimated themselves and in random-effects models, we must also estimate the variance component for heterogeneity, but then we just treated it as a known constant, which isn't quite right either ... but yes, we kind of get the UMVUE if we use inverse-variance weighting if we just squint our eyes very hard and ignore some of these issues.
So, it's efficiency of the estimator that is at stake here, not the unbiasedness itself. But even an unweighted average will often not be a whole lot less efficient than using an inverse-variance weighted average, especially in random-effects models and when the amount of heterogeneity is large (in which case the usual weighting scheme leads to almost uniform weights anyway!). But even in fixed-effects models or with little heterogeneity, the difference often isn't overwhelming.
And as you mention, one can also easily consider other weighting schemes, such as weighting by sample size or some function thereof, but again this is just an attempt to get something close to the inverse-variance weights (since the sampling variances are, to a large extent, determined by the sample size of a study).
But really, one can and should 'decouple' the issue of weights and variances altogether. They are really two separate pieces that one has to think about. But that's just not how things are typically presented in the literature.
However, the point here is that you really need to think about both. Yes, you can take an unweighted average as your combined estimate and that would, in essence, be a meta-analysis, but once you want to start doing inferences based on that combined estimate (e.g., conduct a hypothesis test, construct a confidence interval), you need to know the sampling variances (and the amount of heterogeneity). Think about it this way: If you combine a bunch of small (and/or very heterogeneous) studies, your point estimate is going to be a whole lot less precise than if you combine the same number of very large (and/or homogeneous) studies -- regardless of how you weighted your estimates when calculating the combined value.
Actually, there are even some ways around not knowing the sampling variances (and amount of heterogeneity) when we start doing inferential statistics. One can consider methods based on resampling (e.g., bootstrapping, permutation testing) or methods that yield consistent standard errors for the combined estimate even when we misspecify parts of the model -- but how well these approaches may work needs to be carefully evaluated on a case-by-case basis.
