Is it true that SPF products get into the bloodstream?

Yes, it is. Is it harmful? No clear evidence.
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A 2019 study found that older generation chemical filters (avobenzone, octocrylene, oxybenzone, and mexoryl) can enter the bloodstream. But it’s all ambiguous.
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In a new 2020 study, 48 people applied sunscreen with a chemical filter daily to almost the entire body. After 21 days, chemical filters were found in their blood plasma in concentrations above the acceptable threshold.
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Avobenzone penetrated into the blood more than other filters, three others (homosalate, octinoxate, and octisalate) also penetrate and concentrate at values above the threshold.
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However, don’t panic. There is no evidence that this concentration of UV filters in the body is harmful, which is also emphasized by the authors of the study. Yes, filters do get into the blood, but their protective function (from UV rays) is more obvious than potential harm.
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If you are concerned about chemical filters or know that your body may react negatively to them, use zinc oxide and titanium dioxide (physical filters), which are safe even for hypersensitive skin.