Learn English – Does “Years of experience” take a singular verb

First, you're confusing the issue a bit with a misuse of "a record" in the adjectival sense. "There was a record of 28,000 runners participating" technically¹ means that documentation existed for this number of runners – perhaps there were more runners, but the remainder were not documented. Here, the subject is "record", so the verb is singular. If you're trying to unambiguously remark on the fact that 28,000 is a large number, you need to leave out the "of": "There were a record 28,000 runners participating." Here, the subject is "runners", so the verb is plural.

^{1 In practice, it's an ambiguous statement, because lots of people use "a record of" incorrectly.}

In the general case of "a [x] of [y]", the verb should agree with [x], unless it's one of the special cases (e.g. a lot of) where [x] itself is sorta-kinda plural in meaning, and so can take a plural verb. To figure it out, you can remove the extra words and see if the result sounds correct.

• There was a record of these statements. (There was a record.)
• The use of apples instead of pears was unexpected. (The use was unexpected.)
• It was a time of mass uprisings and protests. (It was a time.)
• There were a lot of people present. (There were a lot.)
• A variety of options were available. (A variety were available.)

I don't think you can categorically identify either version as "incorrect".

As OP says, superficially "We've got years of experience that keep us safe" seems correct, since the subject of the verb keep is the plural noun phrase years of experience.

But I personally have no problem parsing it as experience that keeps us safe. The fact that this experience is qualified by years of seems to me no different to if it had been qualified by plenty of, or a great deal of. It's the singular experience, not the plural years, that helps.

In short, both seem "defensible" to me, but I find the singular verb form flows better.