i think every bitcoin software whether its SPV, SW, LN or core all check every transaction before saving to file or relaying.
SPV doesn't check signatures.
relying on someone else to do checks, to trust them when they say everything is ok, and only double check when they say is wrong seems a little backwards
That's actually how a good share of the ecosystem presently operates. What did you think everyone runs a full node?
It seems to me there are fundamentals holes in your understanding that prevent you from making any progress in your comprehension of SW.