Most lightweight clients do not validate signatures in Bitcoin. While Simplified Payment Verification (SPV) was introduced in the bitcoin whitepaper as a method of verifying payments without the whole blockchain
It is possible to verify payments without running a full network node
SPV has found most of it's usage on wallets that are unable to allocate the resources for a full validating node. The popular example belongs to the mobile wallet.
I wonder if popular lightweight wallets (like Electrum) nowadays do signature checks? If not, why?
Electrum isn't a standard lightweight wallet. Typical lightweight wallets connect via the bitcoin protocol directly to bitcoin seed servers and communicate directly with bitcoin servers.
Electrum does not do this. Electrum lightweight clients connect to Electrum servers, therefor it doesn't make sense for Electrum lightweight clients to take the burden on validation (when at best they could do a sub par job as SPV isn't perfect and results in plenty of false positives)