I think both views can be compatible.

One can support income inequality, while also supporting a safety net for everyone.

Actually, I think a universal income plus free healthcare is desirable. But I also think rewarding people with more income for their achievements is key to sustain progress.

Hawking is probably concerned because the UK is getting a bit too capitalistic, while ignoring that many decisions are suboptimal for most of its inhabitants. For example, the healthcare system is struggling to cope with more patients and an ever decreasing budget. A subpar transport network makes shipping and commuting inefficient and very contaminant. An artificial limit on housing pushes prices up and extracts rent from the majority to hand it in to the ruling class.