As the error message tells you, the file htaccess.txt in your site's root cannot be written to by PHP. This is an ownership / permissions issue, either of the file itself or its containing folder (with a very small chance of it being an open_basedirs issue, though I have not seen any hosting company run by sane people use that kludge in well over ten years).
You said that the folders and files are "writeable". Writeable by whom? Your (S)FTP user which is different than the user PHP runs under? Your hosting control panel which runs as root? Or writeable by the PHP user which is different than the previous two? There's your problem ;)
Please do read very carefully the entire Security Information chapter of the Akeeba Backup documentation to understand how ownership, users, groups, and permissions work on your server.
As for checking the "Ignore most errors" option, it has the effect of Kickstart ignoring most file creation errors (including file write errors due to ownership and permissions). Essentially, you skipped over all files; nothing much was extracted.
Nicholas K. Dionysopoulos
Lead Developer and Director
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