[code-infra] Fix `test_types_next` job

[Janpot]

This is failing every day

• remove the exit 0, we want this to fail, it runs on cron, not on every PR
• remove the cache restoration, nothing was restoring
• remove the strange diffy thing, since nothing is restoring it doesn't make sense
• use correct build script, types build needs stable to have run

[mui-bot]

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Bundle size report

Bundle	Parsed Size	Gzip Size
@mui/material	0B(0.00%)	0B(0.00%)
@mui/lab	0B(0.00%)	0B(0.00%)
@mui/system	0B(0.00%)	0B(0.00%)
@mui/utils	0B(0.00%)	0B(0.00%)

Details of bundle changes

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