Fix SQLite unstable tests

[noisersup]

What should be done?

It seems that some tests from basic_tests.go are still unstable. Every test/subtest pass correctly if ran alone, although if we try to run more of them, there are SQL logic error: no such table: _ferretdb_collections (1) errors.

Some results of investigation here: #3169

Simple stress test that reproduce it: #3195.

Where?

• basic_tests.go
• query_test.go
• update_array_compat_test.go
• update_field_compat_test.go
• update_field_test.go
• SQLite backend

Definition of Done

• all handlers updated;
• unit tests added/updated;
• integration/compatibility tests added/updated;
• spot refactorings done;
• user documentation updated or an issue to create documentation created;
• something else?

[chilagrow]

I can reproduce this locally by running below

$ task test-integration-sqlite TEST_RUN='BenchmarkInsertMany|TestAggregateAddFieldsErrors|TestAggregateCommandMaxTimeMSErrors|TestAggregateCompatCount|TestAggregateCompatGroupCount|TestAggregateCompatGroupExpressionDottedFieldsDocs|TestAggregateCompatMatch|TestAggregateCompatProject|TestAggregateCompatSkip|TestAggregateCompatStages|TestAggregateGroupErrors|TestAggregateSetErrors|TestAggregateVariablesCompatRoot|TestCommandWhatsMyURIConnection|TestCommandsAdministrationCollStats|TestCommandsAdministrationCompatCollStatsWithScale|TestCommandsAdministrationCreateDropListDatabases|TestCommandsAdministrationDBStatsEmpty|TestCommandsAdministrationDataSize|TestCommandsAdministrationKillCursors|TestCommandsAdministrationServerStatus|TestCommandsAdministrationServerStatusStress|TestCommandsAuthenticationSASLStart|TestCommandsDiagnosticGetLog|TestCommandsDiagnosticValidate|TestCommandsFreeMonitoringGetFreeMonitoringStatus|TestCountCommandCompatErrors|TestCreateIndexesCommandCompat|TestCreateIndexesCommandInvalidSpec|TestCreateNestedDocument|TestCreateStress|TestDebugError|TestDeleteCommandCompatNotExistingDatabase|TestDeleteCompat|TestDistinctCommandErrors|TestDotNotation|TestDropIndexesCommandInvalidCollection|TestEmptyKey|TestFindAndModifyCommandErrors|TestFindAndModifyCommentQuery|TestFindAndModifyCompatRemove|TestFindAndModifyCompatUnset|TestFindAndModifyCompatUpdateRename|TestFindAndModifyCompatUpsertSet|TestFindAndModifyNonExistingCollection|TestFindNothing|TestGetMoreCommandConnection|TestGetParameterCommandAuthenticationMechanisms|TestInsertCompat|TestMostCommandsAreCaseSensitive|TestQueryArrayCompatDotNotation|TestQueryArrayCompatSize|TestQueryBitwiseAllClear|TestQueryBitwiseAnySet|TestQueryCommandLimitPushDown|TestQueryComparisonCompatGt|TestQueryComparisonCompatIn|TestQueryComparisonCompatMultipleOperators|TestQueryCompatBatchSize|TestQueryCompatSkip|TestQueryElementCompatElementType'