Would a journal paying money for reviews, with open access and ad-based revenue, work?

Your idea has no chance of success. I'll ignore the part about reviewers being PhD students and postdocs and just look at the publishing model, because that alone is enough to sink the concept.

Ad revenue is not enough to sustain a journal. Just consider: who is willing to pay to advertise in a journal?

• You might be able to get some ad revenue in a print journal, but as a whole, print is declining (maybe ~20% of the journal market at this point and dropping), plus with the most lucrative markets (developed countries) almost fully using electronic articles, you're not going to reach much of an audience that advertisers will be interested in.
• Plus, if you do print your journal, you have printing costs to pay for.
• Even if you go fully electronic, academic journals don't have nearly enough readership to attract advertisers. Just compare how many subscribers you need on a YouTube channel to monetize content (I use YouTube because this is also an advertisement-driven service). YouTube wants one thousand subscribers. Academic articles don't come close to 1,000 readers on average.
• And to top it off, not only are you generating no revenue (diamond open access), you want to pay your reviewers. Where are you going to get the money from? You can beg your editorial board members work for free, you can use a free editorial-management system, you can outsource copyediting and typesetting to the authors, you can ignore marketing your journal, you can strongarm your institution to host your journal's webpage for free ... but paying reviewers involves cold hard cash.

The only way this can work is if you manage to secure renewable, external funding. Depending on how much you're willing to skimp on a journal's operating costs, as well as on how many articles your journal gets, you might not need more than a couple of thousand dollars every year. However even this level of funding is not easy to get. There's a reason why diamond open access journals are rare.

tl; dr: you can try, but expect to fail.

So the deal is that if you drop everything else you are working on to do a review in a week, you earn $10?

I think you need to reconsider that business model, or at least, make the gain worth the disruption. $100 might work better than $10 for "impoverished" grad students, but it's still only chicken feed for a professional.

If somebody offered to pay me $10 to review something, I would put in the amount of effort that $10 was worth. That's probably about 5 minutes of my time - long enough to flick through the pages, and write a couple of sentences saying "awesome" or "garbage". That counts as "a review", yes?

The above intentionally avoids the question of whether the whole idea is ethical from an academic point of view, but if it won't work as a business model, the question of ethics doesn't matter much.

The first thing you need to do when developing a new service is to come up with a realistic business case. What are your fixed and variable costs, and what kind of revenue you can expect?

Fixed costs: at least costs for hosting your website and papers, staff (maybe you initially will run everything yourself in your free time, but if this should become a real business you need to envision a future where you can't do it all on your own anymore), advertisement. All together not too bad (probably).

Variable costs: if you pay a modest 10USD per review, every article will cost you at least 20USD or 30USD to produce. Additionally, you may want to pay editors as well (why would they work for free if the reviewers get payed?). This sounds like peanuts, but it adds up since you are paying this upfront and for every submitted (not only accepted) article, and even for the ones that get viewed less than a 100 times in their lifetime. One journal that I am in the Editorial Board for accepts about 20% of submissions, so your unit costs of an accepted article may be in the range of 100USD or more.

Expected revenue: it is unclear to me if you want to put ads on the website, on the papers, or both. In any case, advertisement revenues live from large numbers of views. Nobody pays you to advertise contents that a few hundred people look at every month. In journals such as PeerJ you can figure out how many views articles typically get, while data from platforms such as Youtube can give you an impression of how many views per paper you will need to offset the variable costs. However, do take into account that views follow a long tail distribution - most content basically nobody ever looks at, so your high runners need to make up for it. This is true for all platforms - Youtube does not earn money with most videos, but when it does, it often makes a lot of money. It is unclear to me how many scientific articles you can expect that would hold the same mass market appeal.

All in all, I am not buying the business case based on this napkin calculation, but it's certainly in the realm of possibility that it could work out financially. You will need to do your own estimations.

That said, you also have some other considerations (which may also sink your business even if the monetary side would in theory work out):

• Bootstrapping a journal outside of the big, established organizations (Springer, ACM, IEEE, ...) is hard. Authors today live and die based on reputation of the journals that they publish in, and your new journal will need to fight to be seen as serious. That it also operates differently will be a strike against it in this dimension. I assume that you hope that your fast reviews will provide sufficient incentives for authors, but I am not buying it - fast reviews are not typically considered to correlate with quality, and neither is that the reviews are done by PhD students and postdocs. You personally may be convinced that PhD students write better reviews than senior academics, but the larger academic world is unlikely to agree. If you want your business to succeed, the community needs to buy into your premise.
• Another aspect that may be held against your journal is that your journal has somewhat unfortunate incentives: it is better for you to accept than to reject (given that rejected articles incur costs but never lead to revenue), and it is even better to accept highly controversial (but wrong) articles that get viewed a lot. Even if your journal never gives in to these temptations, I foresee that there will be a certain level of mistrust hanging over the scientific trustworthiness of your journal.