BUG: to_html misses truncation indicators (...) when index=False
- [x] closes #15019
- [x] closes #22783
- [x] tests added / passed
- [x] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [x] whatsnew entry

Hello @simonjayhawkins! Thanks for submitting the PR.

 - There are no PEP8 issues in the file [`pandas/io/formats/html.py`](https://github.com/simonjayhawkins/pandas/blob/a7b24248a75838ede41e50a1c073c9f03e33426b/pandas/io/formats/html.py) !

 - There are no PEP8 issues in the file [`pandas/tests/io/formats/conftest.py`](https://github.com/simonjayhawkins/pandas/blob/a7b24248a75838ede41e50a1c073c9f03e33426b/pandas/tests/io/formats/conftest.py) !

 - There are no PEP8 issues in the file [`pandas/tests/io/formats/test_to_html.py`](https://github.com/simonjayhawkins/pandas/blob/a7b24248a75838ede41e50a1c073c9f03e33426b/pandas/tests/io/formats/test_to_html.py) !

 @simonjayhawkins I know you have a few overlapping PRs open at this point - is this the one you think should go in first? Want to make sure I prioritize review accordingly @WillAyd : Yes Will. Although not printing the column names #22747 is a probably a simpler fix and could go first. I've not raised a PR for that yet, as there's been no affirmation it's an issue. @WillAyd : Will, the refactoring PR, #22726 is independent to keep style changes and bug fixes separate. So there is no reason that #22726 can't be progressed in parallel. # [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/22786?src=pr&el=h1) Report
> Merging [#22786](https://codecov.io/gh/pandas-dev/pandas/pull/22786?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/c55057f79b0cbe47cf73f0fd8141e6c84f1e7370?src=pr&el=desc) will **increase** coverage by `<.01%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/22786/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/22786?src=pr&el=tree)

```diff
@@            Coverage Diff             @@
##           master   #22786      +/-   ##
==========================================
+ Coverage   92.24%   92.24%   +<.01%     
==========================================
  Files         161      161              
  Lines       51336    51338       +2     
==========================================
+ Hits        47357    47359       +2     
  Misses       3979     3979
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.64% <100%> (ø)` | :arrow_up: |
| #single | `42.34% <0%> (-0.01%)` | :arrow_down: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/22786?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/io/formats/html.py](https://codecov.io/gh/pandas-dev/pandas/pull/22786/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL2Zvcm1hdHMvaHRtbC5weQ==) | `93.18% <100%> (+0.04%)` | :arrow_up: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/22786?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/22786?src=pr&el=footer). Last update [c55057f...15bd352](https://codecov.io/gh/pandas-dev/pandas/pull/22786?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).
 @jreback : This PR is currently blocking #22655, hence the overlap. I've marked it as WIP for the moment. can you merge master @WillAyd can you have another look here @WillAyd @jreback I think all your comments have been addressed, right? @simonjayhawkins code lgtm. Can you resolve conflicts? Can merge thereafter on green