core: Something is blocking Home Assistant from wrapping up the start up

The problem

Since 2021-04 update, I am not able to get HASS fully started. Something is slowing the system, Automations won’t start. I have been using 2021-03 as the last usable version. After update (for both .04 and .05), I can access the UI from the cloud, whereas the local connection fails to load at all.
The logs have multiple CORE-originating error messages that do not mean much (to me), for example, there are tens of WARNING (MainThread) [homeassistant.components.mqtt] No ACK from MQTT server in 10 seconds (mid: 21) in a row, as well as other types of warning that something is taking a long time to connect: WARNING (MainThread) [homeassistant.helpers.entity] Update of weather.dark_sky is taking over 10 seconds

I have a couple of custom integrations, tried disabling those before updating (2021.04 update), but this did not help.

What is version of Home Assistant Core has the issue?

core-2021.5.0

What was the last working version of Home Assistant Core?

No response

What type of installation are you running?

Home Assistant OS

Integration causing the issue

No response

Link to integration documentation on our website

No response

Example YAML snippet

No response

Anything in the logs that might be useful for us?

2021-05-06 11:46:31 WARNING (MainThread) [homeassistant.core] Something is blocking Home Assistant from wrapping up the start up phase. We're going to continue anyway. Please report the following info at https://github.com/home-assistant/core/issues: zha, history, my, cast, input_number, updater, fan.zha, vacuum, influxdb, tasmota, zone, notify.ios, persistent_notification, ozw, remote.harmony, discovery, zeroconf, ipp, ssdp, number.mqtt, input_datetime, songpal, climate.ozw, stream, light.ozw, light.zha, calendar, frontend, sensor.ozw, tag, analytics, input_text, lovelace, lock.zha, media_source, cover.ozw, scene, system_log, sensor, sonos, number, cover.shelly, fan.tasmota, script, esphome, light.shelly, media_player.sonos, openmediavault, template, switch.shelly, device_automation, sun, shelly, device_tracker, sensor.zha, homeassistant, cover.template, sensor.garbage_collection, binary_sensor.updater, binary_sensor.openmediavault, tts.cloud, scene.homeassistant, websocket_api, fan, sensor.template, vacuum.xiaomi_miio, binary_sensor.mqtt, calendar.garbage_collection, notify.pushbullet, input_boolean, kodi, sensor.hacs, tts, garbage_collection, onboarding, device_tracker.zha, fan.mqtt, hacs, sensor.mqtt, media_player, switch.harmony, binary_sensor.esphome, sensor.shelly, device_tracker.unifi, number.zha, climate, blueprint, sensor.esphome, fan.esphome, binary_sensor.cloud, dhcp, person, sensor.sonos, search, fan.ozw, weather, sensor.openmediavault, map, harmony, image, ios, sensor.hassio, config, alexa, remote, tts.google_translate, cover, cloud, alert, http, switch.ozw, stt, cover.zha, lock, notify, sensor.darksky, hassio, weather.darksky, sensor.tasmota, climate.zha, binary_sensor.hassio, binary_sensor.tasmota, switch.zha, auth, panel_iframe, counter, light.esphome, google_assistant, mobile_app, sensor.history_stats, binary_sensor.mobile_app, sensor.unifi, light, binary_sensor.shelly, notify.mobile_app, alarm_control_panel.zha, switch, mqtt, default_config, lock.ozw, switch.esphome, sensor.mobile_app, logbook, switch.tasmota, conversation, switch.command_line, sensor.ios, binary_sensor.ozw, device_tracker.mobile_app, timer, sonos_alarm, trace, binary_sensor.sonos, sensor.modbus, automation, binary_sensor.zha, switch.unifi, api, media_player.cast, alarm_control_panel, switch.sonos_alarm, system_health, switch.mqtt, recorder, webhook, binary_sensor, cover.mqtt, media_player.kodi, modbus, cover.tasmota, input_select, light.mqtt, sensor.filesize, light.tasmota, xiaomi_miio, unifi

Additional information

No response